Euler has design of a homepage and there is a link back to the overview for each solution.
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"cell_type": "markdown",
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"source": [
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"# Euler Problem 13\n",
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"\n",
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"Work out the first ten digits of the sum of the following one-hundred 50-digit numbers.\n",
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"\n",
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"~~~\n",
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"completion_date": "Sun, 31 Aug 2014, 19:03",
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"display_name": "Python 3",
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"name": "python",
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"version": "3.6.3"
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"version": "3.6.5"
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},
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"tags": [
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"brute force",
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<!DOCTYPE html>
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<title>EulerProblem001</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
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<h1 id="Euler-Problem-1">Euler Problem 1<a class="anchor-link" href="#Euler-Problem-1">¶</a></h1><p>If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.</p>
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<h1 id="Euler-Problem-1">Euler Problem 1<a class="anchor-link" href="#Euler-Problem-1">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.</p>
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<p>Find the sum of all the multiples of 3 or 5 below 1000.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_sum_of_natural_dividable_by_3_and_5_below</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
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<span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="k">if</span> <span class="n">x</span> <span class="o">%</span> <span class="mi">3</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">x</span> <span class="o">%</span> <span class="mi">5</span> <span class="o">==</span> <span class="mi">0</span><span class="p">])</span>
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<p>Test example provided in problem statement:</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">assert</span><span class="p">(</span><span class="n">get_sum_of_natural_dividable_by_3_and_5_below</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="mi">23</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">get_sum_of_natural_dividable_by_3_and_5_below</span><span class="p">(</span><span class="mi">1000</span><span class="p">))</span>
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<pre>233168
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<title>EulerProblem002</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
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/*!
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<h1 id="Euler-Problem-2">Euler Problem 2<a class="anchor-link" href="#Euler-Problem-2">¶</a></h1><p>Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:</p>
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<h1 id="Euler-Problem-2">Euler Problem 2<a class="anchor-link" href="#Euler-Problem-2">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:</p>
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<p>1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...</p>
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<p>By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">fibonacci_generator</span><span class="p">():</span>
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<span class="n">a</span> <span class="o">=</span> <span class="mi">0</span>
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<span class="n">b</span> <span class="o">=</span> <span class="mi">1</span>
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<span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
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<p>Test fibonacci sequence generator by comparing the result to the example in the task statement.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">i</span> <span class="o">=</span> <span class="n">fibonacci_generator</span><span class="p">()</span>
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<span class="n">fib_10</span> <span class="o">=</span> <span class="p">[</span><span class="nb">next</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)]</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">fib_10</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">21</span><span class="p">,</span> <span class="mi">34</span><span class="p">,</span> <span class="mi">55</span><span class="p">,</span> <span class="mi">89</span><span class="p">,])</span>
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<p>Greate a function which returns all even-valued fibonacci values smaller or equal to four million.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_even_fibonaccis_smaller_or_equal_four_million</span><span class="p">():</span>
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<span class="n">r</span> <span class="o">=</span> <span class="p">[]</span>
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<span class="n">i</span> <span class="o">=</span> <span class="n">fibonacci_generator</span><span class="p">()</span>
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<span class="n">current_value</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
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<span class="k">return</span> <span class="n">r</span>
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<p>Calculate the solution.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f</span> <span class="o">=</span> <span class="n">get_even_fibonaccis_smaller_or_equal_four_million</span><span class="p">()</span>
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<span class="nb">print</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">f</span><span class="p">))</span>
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<pre>4613732
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<p>I looked at the solutions in the forum and I kind of forgot about simple straight forward approaches. There is no need to create a list and the sum it up. Instead I can simply increment a counter which will be much faster, but less readable maybe.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">r</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span>
|
||||
|
||||
<span class="k">while</span> <span class="n">b</span> <span class="o"><=</span> <span class="mi">4000000</span><span class="p">:</span>
|
||||
|
|
@ -11919,35 +11894,22 @@ div#notebook {
|
|||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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||||
|
||||
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|
||||
|
||||
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|
||||
|
||||
|
||||
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|
||||
<pre>4613732
|
||||
</pre>
|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
|
||||
|
||||
|
||||
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||||
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|
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|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem003</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
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||||
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|
||||
/*!
|
||||
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|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
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||||
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|
|||
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|
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|
||||
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<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
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|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
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|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,18 +11762,16 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
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||||
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|
||||
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|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
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|
||||
<h1 id="Euler-Problem-003">Euler Problem 003<a class="anchor-link" href="#Euler-Problem-003">¶</a></h1><p>The prime factors of 13195 are 5, 7, 13 and 29.</p>
|
||||
<h1 id="Euler-Problem-003">Euler Problem 003<a class="anchor-link" href="#Euler-Problem-003">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The prime factors of 13195 are 5, 7, 13 and 29.</p>
|
||||
<p>What is the largest prime factor of the number 600851475143?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11786,15 +11780,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
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|
||||
<p>We start by writing a function which calculates all primes till a certain value using the Sieve of Eratosthenes.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
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|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_primes_smaller</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
|
||||
<span class="n">primes</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">prospects</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">number</span><span class="p">)]</span>
|
||||
|
|
@ -11807,26 +11800,23 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">get_primes_smaller</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">==</span> <span class="p">[])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_primes_smaller</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Now we create a function which does prime factorization. It is very important that we only test primes smaller than the squre root. Otherwise the complexity becomes too big.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">math</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">get_prime_factors</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
|
||||
|
|
@ -11848,26 +11838,23 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">get_prime_factors</span><span class="p">(</span><span class="mi">88</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">11</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_prime_factors</span><span class="p">(</span><span class="mi">13195</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">29</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
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|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Now we can go ahead an brute force the solution.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_largest_prime</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="n">get_prime_factors</span><span class="p">(</span><span class="n">number</span><span class="p">)[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||||
|
||||
|
|
@ -11875,44 +11862,34 @@ div#notebook {
|
|||
<span class="c1">#print(get_largest_prime(600851475143))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="mi">6857</span><span class="p">)</span> <span class="c1"># computed the previously but remove it so that we can reexecute the complete kernel</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
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|
||||
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||||
|
||||
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|
||||
|
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|
||||
|
||||
|
||||
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|
||||
<pre>6857
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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||||
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|
||||
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|
||||
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|
||||
<p>Okay, actually we can brute force, but it is really slow. A better solution is to write a prime number generator which calculates the next number on demand.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
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|
||||
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|
||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_prime</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">smaller_primes</span><span class="p">):</span>
|
||||
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">smaller_primes</span><span class="p">:</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="n">s</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
|
|
@ -11947,44 +11924,30 @@ div#notebook {
|
|||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">get_largest_prime</span><span class="p">(</span><span class="mi">600851475143</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>6857
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Here we go. Obviously much better than precalculation primes that we never need.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
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|
||||
</div>
|
||||
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|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem004</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,18 +11762,16 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<h1 id="Euler-Problem-4">Euler Problem 4<a class="anchor-link" href="#Euler-Problem-4">¶</a></h1><p>A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.</p>
|
||||
<h1 id="Euler-Problem-4">Euler Problem 4<a class="anchor-link" href="#Euler-Problem-4">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.</p>
|
||||
<p>Find the largest palindrome made from the product of two 3-digit numbers.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11786,15 +11780,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>No big magic going on in this solution. We apply brute force. The only optimization we take is starting from the back so that the first palindrome should be the largest. However, we first implement a simple function to check whether an integer is a palindrome.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_palindrome</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">)</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
|
|
@ -11811,67 +11804,54 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">is_palindrome</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_palindrome</span><span class="p">(</span><span class="mi">908</span><span class="p">)</span> <span class="ow">is</span> <span class="kc">False</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
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<p>Get potential solutions. Generating a list comprehension which sorts the products in decreasing order is actually not that easys. This is because if we use two ranges it is not straight foward to decide which product is smaller or bigger. In the following example. 9 times 7 is still bigger than 8 times 8, but 9 times 6 is not anymore. So let's go for brute force the hard way.</p>
|
||||
|
||||
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|
||||
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|
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|
||||
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">pairs</span> <span class="o">=</span> <span class="p">((</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">pairs</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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<pre>[(9, 9), (9, 8), (9, 7), (9, 6), (9, 5), (9, 4), (9, 3), (9, 2), (9, 1), (8, 8), (8, 7), (8, 6), (8, 5), (8, 4), (8, 3), (8, 2), (8, 1), (7, 7), (7, 6), (7, 5), (7, 4), (7, 3), (7, 2), (7, 1), (6, 6), (6, 5), (6, 4), (6, 3), (6, 2), (6, 1), (5, 5), (5, 4), (5, 3), (5, 2), (5, 1), (4, 4), (4, 3), (4, 2), (4, 1), (3, 3), (3, 2), (3, 1), (2, 2), (2, 1), (1, 1)]
|
||||
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|
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||||
<p>We iterate over all 3-digit pairs. Two avoid duplicates the second range starts at the current value of the first range. If the new product is greater than the old one and a palindrome we have found a new result.</p>
|
||||
|
||||
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||||
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||||
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">timeit</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">brute_force</span><span class="p">():</span>
|
||||
|
|
@ -11886,45 +11866,35 @@ div#notebook {
|
|||
<span class="nb">print</span><span class="p">(</span><span class="n">timeit</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">brute_force</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="mi">100</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">brute_force</span><span class="p">())</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
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<pre>2.844880788875627
|
||||
906609
|
||||
</pre>
|
||||
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|
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<p>To do some more optimization we can simply break the loops if the square of the outer loop is smaller than the current result. If we find the solution quiet early this will prevent iterating through a lot of values which cannot yield a better result.</p>
|
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">brute_force</span><span class="p">():</span>
|
||||
<span class="n">r</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
<span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">999</span><span class="p">,</span> <span class="mi">99</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
||||
|
|
@ -11939,36 +11909,23 @@ div#notebook {
|
|||
<span class="nb">print</span><span class="p">(</span><span class="n">timeit</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">brute_force</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="mi">100</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">brute_force</span><span class="p">())</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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||||
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||||
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||||
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||||
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|
||||
<pre>0.562125718678324
|
||||
906609
|
||||
</pre>
|
||||
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|
||||
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|
||||
|
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|
||||
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|
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||||
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|
||||
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||||
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|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
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||||
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||||
<html>
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||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem005</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
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<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
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/*!
|
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*
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@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
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|
||||
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|
||||
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@ -11741,15 +11739,13 @@ div#notebook {
|
|||
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|
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|
||||
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|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
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|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
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||||
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||||
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|
||||
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||||
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|
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||||
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|
||||
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|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,18 +11762,16 @@ div#notebook {
|
|||
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|
||||
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|
||||
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|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
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|
||||
|
||||
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|
||||
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||||
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||||
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||||
<h1 id="Euler-Problem">Euler Problem<a class="anchor-link" href="#Euler-Problem">¶</a></h1><p>2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.</p>
|
||||
<h1 id="Euler-Problem">Euler Problem<a class="anchor-link" href="#Euler-Problem">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.</p>
|
||||
<p>What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?</p>
|
||||
|
||||
</div>
|
||||
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|
||||
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|
||||
|
|
@ -11786,15 +11780,14 @@ div#notebook {
|
|||
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|
||||
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|
||||
<p>My easiest guess is to multiply all prime numbers till the number.</p>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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||||
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||||
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_primes_smaller</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
|
||||
<span class="n">primes</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">prospects</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">number</span><span class="p">)]</span>
|
||||
|
|
@ -11804,17 +11797,15 @@ div#notebook {
|
|||
<span class="n">primes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">primes</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
<div class="input">
|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">operator</span> <span class="k">import</span> <span class="n">mul</span>
|
||||
<span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">reduce</span>
|
||||
|
||||
|
|
@ -11824,44 +11815,34 @@ div#notebook {
|
|||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">get_number_which_is_divisible_by_all_numbers_from_one_to</span><span class="p">(</span><span class="mi">10</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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||||
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||||
<pre>210
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
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|
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||||
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||||
<p>That obviously didn't work. The reason is that the same prime can occur multiple times in the factorization of a divisor. For example $2^{3} = 8$. We can always brute force of course. We do a smart brute force and only check multiples from the product of primes because this factor must be part of the solution.</p>
|
||||
|
||||
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||||
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|
||||
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|
||||
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||||
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||||
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||||
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||||
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||||
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||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_divisible_by_numbers_smaller_or_equal</span><span class="p">(</span><span class="n">number</span><span class="p">,</span> <span class="n">maximum_number</span><span class="p">):</span>
|
||||
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">maximum_number</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="n">number</span> <span class="o">%</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
|
||||
|
|
@ -11880,35 +11861,22 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">get_number_which_is_divisible_by_all_numbers_from_one_to</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2520</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">get_number_which_is_divisible_by_all_numbers_from_one_to</span><span class="p">(</span><span class="mi">20</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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||||
|
||||
|
||||
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|
||||
<pre>232792560
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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||||
|
||||
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||||
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||||
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|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem006</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,22 +11762,20 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-6">Euler Problem 6<a class="anchor-link" href="#Euler-Problem-6">¶</a></h1><p>The sum of the squares of the first ten natural numbers is,</p>
|
||||
<h1 id="Euler-Problem-6">Euler Problem 6<a class="anchor-link" href="#Euler-Problem-6">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The sum of the squares of the first ten natural numbers is,</p>
|
||||
<p>$1^2 + 2^2 + ... + 10^2 = 385$</p>
|
||||
<p>The square of the sum of the first ten natural numbers is,</p>
|
||||
<p>$(1 + 2 + ... + 10)^2 = 55^2 = 3025$</p>
|
||||
<p>Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is $3025 − 385 = 2640$.</p>
|
||||
<p>Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11790,74 +11784,57 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, this is as straightforward as it can get.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">101</span><span class="p">)])</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">101</span><span class="p">)])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">25164150</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>25164150
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>General solution.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">diff_between_sum_of_squares_and_square_sum</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">diff_between_sum_of_squares_and_square_sum</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span> <span class="o">==</span> <span class="mi">25164150</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">diff_between_sum_of_squares_and_square_sum</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2640</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem007</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,18 +11762,16 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-7">Euler Problem 7<a class="anchor-link" href="#Euler-Problem-7">¶</a></h1><p>By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.</p>
|
||||
<h1 id="Euler-Problem-7">Euler Problem 7<a class="anchor-link" href="#Euler-Problem-7">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.</p>
|
||||
<p>What is the 10 001st prime number?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11786,15 +11780,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We reuse our prime functions and use a trick from <a href="https://stackoverflow.com/questions/2300756/get-the-nth-item-of-a-generator-in-python">stackoverflow</a> to get the nth element.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [7]:</div>
|
||||
<div class="prompt input_prompt">In [7]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">itertools</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">is_prime</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">smaller_primes</span><span class="p">):</span>
|
||||
|
|
@ -11822,35 +11815,22 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">get_nth_prime</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span> <span class="o">==</span> <span class="mi">13</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">get_nth_prime</span><span class="p">(</span><span class="mi">10001</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>104743
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem008</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,17 +11762,15 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-8">Euler Problem 8<a class="anchor-link" href="#Euler-Problem-8">¶</a></h1><p>The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832.</p>
|
||||
|
||||
<h1 id="Euler-Problem-8">Euler Problem 8<a class="anchor-link" href="#Euler-Problem-8">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The four adjacent digits in the 1000-digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832.</p>
|
||||
<pre><code>73167176531330624919225119674426574742355349194934
|
||||
96983520312774506326239578318016984801869478851843
|
||||
85861560789112949495459501737958331952853208805511
|
||||
|
|
@ -11798,7 +11792,6 @@ div#notebook {
|
|||
05886116467109405077541002256983155200055935729725
|
||||
71636269561882670428252483600823257530420752963450</code></pre>
|
||||
<p>Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11807,20 +11800,19 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We need the number as a list of integers first.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">remove_newlines</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
|
||||
<span class="kn">import</span> <span class="nn">re</span>
|
||||
<span class="k">return</span> <span class="n">re</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="sa">r</span><span class="s1">'\n'</span><span class="p">,</span> <span class="s1">''</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">re</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="sa">r</span><span class="s1">'\n'</span><span class="p">,</span> <span class="s1">''</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
|
||||
|
||||
<span class="n">digits_string</span> <span class="o">=</span> <span class="s2">"""73167176531330624919225119674426574742355349194934</span>
|
||||
<span class="n">digits_string</span> <span class="o">=</span> <span class="s2">"""73167176531330624919225119674426574742355349194934</span>
|
||||
<span class="s2">96983520312774506326239578318016984801869478851843</span>
|
||||
<span class="s2">85861560789112949495459501737958331952853208805511</span>
|
||||
<span class="s2">12540698747158523863050715693290963295227443043557</span>
|
||||
|
|
@ -11839,30 +11831,27 @@ div#notebook {
|
|||
<span class="s2">07198403850962455444362981230987879927244284909188</span>
|
||||
<span class="s2">84580156166097919133875499200524063689912560717606</span>
|
||||
<span class="s2">05886116467109405077541002256983155200055935729725</span>
|
||||
<span class="s2">71636269561882670428252483600823257530420752963450"""</span>
|
||||
<span class="s2">71636269561882670428252483600823257530420752963450"""</span>
|
||||
|
||||
<span class="n">digits</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">d</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">remove_newlines</span><span class="p">(</span><span class="n">digits_string</span><span class="p">)]</span>
|
||||
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|
||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
<p>Then we can use slicing to do a brute force.</p>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">product</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
|
||||
<span class="kn">from</span> <span class="nn">operator</span> <span class="k">import</span> <span class="n">mul</span>
|
||||
<span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">reduce</span>
|
||||
|
|
@ -11875,35 +11864,22 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">get_largest_product_of_n_digits</span><span class="p">(</span><span class="n">digits</span><span class="p">,</span> <span class="mi">13</span><span class="p">)</span> <span class="o">==</span> <span class="mi">23514624000</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">get_largest_product_of_n_digits</span><span class="p">(</span><span class="n">digits</span><span class="p">,</span> <span class="mi">13</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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|
||||
<pre>23514624000
|
||||
</pre>
|
||||
</div>
|
||||
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|
||||
|
||||
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|
||||
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||||
|
||||
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|
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|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
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<html>
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||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem009</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
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||||
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||||
/*!
|
||||
*
|
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@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
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|
||||
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|
||||
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@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
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|
||||
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||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
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|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,21 +11762,19 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
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|
||||
|
||||
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|
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||||
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||||
<h1 id="Euler-Problem-9">Euler Problem 9<a class="anchor-link" href="#Euler-Problem-9">¶</a></h1><p>A Pythagorean triplet is a set of three natural numbers, $a < b < c$, for which,</p>
|
||||
<h1 id="Euler-Problem-9">Euler Problem 9<a class="anchor-link" href="#Euler-Problem-9">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>A Pythagorean triplet is a set of three natural numbers, $a < b < c$, for which,</p>
|
||||
<p>$a^2 + b^2 = c^2$</p>
|
||||
<p>For example, $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.</p>
|
||||
<p>There exists exactly one Pythagorean triplet for which $a + b + c = 1000$.
|
||||
Find the product $abc$.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11789,15 +11783,14 @@ Find the product $abc$.</p>
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We start bruteforcing even though it feels like there may be a smart solution.</p>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
<div class="cell border-box-sizing code_cell rendered">
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||||
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||||
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||||
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||||
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||||
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||||
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||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">timeit</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">brute_force</span><span class="p">():</span>
|
||||
|
|
@ -11810,45 +11803,35 @@ Find the product $abc$.</p>
|
|||
<span class="nb">print</span><span class="p">(</span><span class="n">timeit</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">brute_force</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="mi">10</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">brute_force</span><span class="p">())</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
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|
||||
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|
||||
|
||||
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||||
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|
||||
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||||
|
||||
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|
||||
<pre>66.96907186399949
|
||||
31875000
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
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|
||||
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|
||||
|
||||
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||||
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||||
<p>Let's do some optimization by cancelling the loops when we exceed the boundaries. Actually, I have also realized that choosing 251 and 501 is a little bit arbitrary. For example if a, b, c where something like $332, 333, 335$ that could be a solution and the first range was too low. Then again, we would have realized that as soon as we get back None, so it is okay.</p>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
<div class="cell border-box-sizing code_cell rendered">
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||||
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||||
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||||
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||||
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||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">brute_force</span><span class="p">():</span>
|
||||
<span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">251</span><span class="p">):</span>
|
||||
<span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">501</span><span class="p">):</span>
|
||||
|
|
@ -11861,45 +11844,35 @@ Find the product $abc$.</p>
|
|||
<span class="nb">print</span><span class="p">(</span><span class="n">timeit</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">brute_force</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="mi">10</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">brute_force</span><span class="p">())</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
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|
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|
||||
|
||||
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|
||||
|
||||
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|
||||
|
||||
|
||||
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|
||||
<pre>64.4656584559998
|
||||
31875000
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
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|
||||
<div class="inner_cell">
|
||||
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|
||||
<p>Big time save. Kappa. Okay, I am stupid. If I have a and b I can calculate c and check if it is a solution.</p>
|
||||
|
||||
</div>
|
||||
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|
||||
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|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">smart_brute_force</span><span class="p">():</span>
|
||||
<span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">251</span><span class="p">):</span>
|
||||
<span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">501</span><span class="p">):</span>
|
||||
|
|
@ -11910,49 +11883,34 @@ Find the product $abc$.</p>
|
|||
<span class="nb">print</span><span class="p">(</span><span class="n">timeit</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">smart_brute_force</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="mi">10</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">smart_brute_force</span><span class="p">())</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
|
||||
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|
||||
|
||||
|
||||
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|
||||
<pre>0.22822808900036762
|
||||
31875000
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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||||
|
||||
|
||||
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||||
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||||
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|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem010</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,18 +11762,16 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
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|
||||
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|
||||
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|
||||
<h1 id="Euler-Problem-10">Euler Problem 10<a class="anchor-link" href="#Euler-Problem-10">¶</a></h1><p>The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.</p>
|
||||
<h1 id="Euler-Problem-10">Euler Problem 10<a class="anchor-link" href="#Euler-Problem-10">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.</p>
|
||||
<p>Find the sum of all the primes below two million.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11786,15 +11780,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, reuse prime generator from 7 and go.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">timeit</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">is_prime</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">smaller_primes</span><span class="p">):</span>
|
||||
|
|
@ -11828,45 +11821,35 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">brute_force</span><span class="p">()</span> <span class="o">==</span> <span class="mi">142913828922</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">brute_force</span><span class="p">())</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
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||||
|
||||
<div class="output_area">
|
||||
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||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>61.8695481220002
|
||||
142913828922
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
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||||
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||||
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||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, here it may actually be way smarter to use a sieve. We implent it because the old one was shitty. Okay, I am actually interested in the time difference. So we use the old one first and then implement the optimization.</p>
|
||||
|
||||
</div>
|
||||
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|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
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||||
<div class="input">
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||||
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||||
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||||
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<div class="input_area">
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||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_primes_smaller</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
|
||||
<span class="n">primes</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">prospects</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">number</span><span class="p">)]</span>
|
||||
|
|
@ -11882,26 +11865,23 @@ div#notebook {
|
|||
<span class="c1">#print(timeit.timeit(brute_force, number=1))</span>
|
||||
<span class="c1">#print(brute_force())</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
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||||
</div>
|
||||
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||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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<div class="inner_cell">
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||||
<div class="text_cell_render border-box-sizing rendered_html">
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||||
<p>This did not even terminate. We optimize the sieve by stopping when $p^2 > number$.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
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||||
<div class="input">
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<div class="prompt input_prompt">In [3]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">sieve_of_eratosthenes</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
|
||||
<span class="n">primes</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">prospects</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">number</span><span class="p">)]</span>
|
||||
|
|
@ -11921,45 +11901,31 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">brute_force</span><span class="p">()</span> <span class="o">==</span> <span class="mi">142913828922</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">brute_force</span><span class="p">())</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
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|
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||||
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<div class="prompt"></div>
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||||
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||||
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||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>74.5315607270004
|
||||
142913828922
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
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|
||||
</div>
|
||||
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||||
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||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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||||
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||||
<div class="inner_cell">
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||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, I honestly did not expect this to be slower than our generator. Gotta keep that in mind for future prime related problems.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
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|
||||
</div>
|
||||
</div>
|
||||
</body>
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||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem011</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
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||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,17 +11762,15 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
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|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-11">Euler Problem 11<a class="anchor-link" href="#Euler-Problem-11">¶</a></h1><p>In the 20×20 grid below, four numbers along a diagonal line have been marked in red.</p>
|
||||
|
||||
<h1 id="Euler-Problem-11">Euler Problem 11<a class="anchor-link" href="#Euler-Problem-11">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>In the 20×20 grid below, four numbers along a diagonal line have been marked in red.</p>
|
||||
<pre><code>08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
|
||||
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
|
||||
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
|
||||
|
|
@ -11799,7 +11793,6 @@ div#notebook {
|
|||
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48</code></pre>
|
||||
<p>The product of these numbers is $26 × 63 × 78 × 14 = 1788696$.</p>
|
||||
<p>What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20×20 grid?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11809,16 +11802,15 @@ div#notebook {
|
|||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We start by parsing the array into three lists of lists (horizontal, diagonal, vertical).</p>
|
||||
<p>97, 17, 79, 11, 89, 44, 38, 94, 78, 0, 62, 54, 58, 4, 27, 53, 36, 1, 1, 1]</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="s2">"""</span>
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="s2">"""</span>
|
||||
<span class="s2">08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08</span>
|
||||
<span class="s2">49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00</span>
|
||||
<span class="s2">81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65</span>
|
||||
|
|
@ -11839,7 +11831,7 @@ div#notebook {
|
|||
<span class="s2">20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16</span>
|
||||
<span class="s2">20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54</span>
|
||||
<span class="s2">01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48</span>
|
||||
<span class="s2">"""</span>
|
||||
<span class="s2">"""</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">get_index</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="n">default</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="n">index</span> <span class="o"><</span> <span class="mi">0</span><span class="p">:</span>
|
||||
|
|
@ -11849,16 +11841,14 @@ div#notebook {
|
|||
<span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">default</span>
|
||||
|
||||
<span class="n">hss</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">h</span><span class="o">.</span><span class="n">split</span><span class="p">()))</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">s</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s1">'</span><span class="se">\n</span><span class="s1">'</span><span class="p">)</span> <span class="k">if</span> <span class="n">h</span><span class="p">]</span>
|
||||
<span class="n">hss</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">h</span><span class="o">.</span><span class="n">split</span><span class="p">()))</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">s</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s1">'</span><span class="se">\n</span><span class="s1">'</span><span class="p">)</span> <span class="k">if</span> <span class="n">h</span><span class="p">]</span>
|
||||
<span class="n">vss</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">vs</span><span class="p">)</span> <span class="k">for</span> <span class="n">vs</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">hss</span><span class="p">)]</span>
|
||||
<span class="c1"># diagonal from top left to bottom right</span>
|
||||
<span class="n">dss</span> <span class="o">=</span> <span class="p">[[</span><span class="n">get_index</span><span class="p">(</span><span class="n">hs</span><span class="p">,</span> <span class="o">-</span><span class="mi">19</span> <span class="o">+</span> <span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">hs</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">hss</span><span class="p">)]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">39</span><span class="p">)]</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
|
|
@ -11866,15 +11856,14 @@ div#notebook {
|
|||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>I really love the way we retrieve the vertical list. (We need the ugly conversion to list because otherwise we cannot concatenate them.) It is at the limit of my mental capabilities to understand why this works. But actually it is quite straight forward. If we provide two lists, zip creates a two-tuple for each field in the lists. If we provide n lists, zip creates a n-tuple for each field in the lists. Easy.</p>
|
||||
<p>Now we create big list of lists and search each one for the greates four product and then get the maximum. Straight forward. We borrow the find product function from problem 8 because I like it.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">xss</span> <span class="o">=</span> <span class="n">hss</span> <span class="o">+</span> <span class="n">vss</span> <span class="o">+</span> <span class="n">dss</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">product</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
|
||||
|
|
@ -11888,26 +11877,23 @@ div#notebook {
|
|||
<span class="n">s</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([</span><span class="n">get_largest_product_of_n_digits</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="n">xss</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">!=</span> <span class="mi">70600674</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, I am actually really dumb. I forgot about the other diagonal. I think I have made the same mistake back when I solved this for the first time. So let's get the missing dss and get the solution.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># diagonal from bottom left to top right</span>
|
||||
<span class="n">hss</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
|
||||
<span class="n">dss</span> <span class="o">=</span> <span class="p">[[</span><span class="n">get_index</span><span class="p">(</span><span class="n">hs</span><span class="p">,</span> <span class="o">-</span><span class="mi">19</span> <span class="o">+</span> <span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">hs</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">hss</span><span class="p">)]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">39</span><span class="p">)]</span>
|
||||
|
|
@ -11916,35 +11902,22 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">70600674</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>70600674
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
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<h1 id="Euler-Problem-12">Euler Problem 12<a class="anchor-link" href="#Euler-Problem-12">¶</a></h1><p>The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be $1 + 2 + 3 + 4 + 5 + 6 + 7 = 28$. The first ten terms would be:</p>
|
||||
<h1 id="Euler-Problem-12">Euler Problem 12<a class="anchor-link" href="#Euler-Problem-12">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be $1 + 2 + 3 + 4 + 5 + 6 + 7 = 28$. The first ten terms would be:</p>
|
||||
<p>$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots$</p>
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<p>Let us list the factors of the first seven triangle numbers:</p>
|
||||
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||||
<pre><code> 1: 1
|
||||
3: 1,3
|
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6: 1,2,3,6
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28: 1,2,4,7,14,28</code></pre>
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<p>We can see that 28 is the first triangle number to have over five divisors.</p>
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<p>What is the value of the first triangle number to have over five hundred divisors?</p>
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<p>Let's try brute forcing.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_divisors</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
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||||
<span class="k">return</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">n</span><span class="p">]</span>
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<span class="c1"># print(t)</span>
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<span class="c1"># break</span>
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</pre></div>
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<p>That failed miserably. We have to come up with something more efficient. I looked at my old Haskell solution which looks like this:</p>
|
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|
||||
<pre><code>divisor_count' x = product . map (succ . length) . group $ prim_factors x</code></pre>
|
||||
<p>Add first I did not understand what it does at all. But the algorithm is as follows:</p>
|
||||
|
||||
<pre><code># get prime factors, for example for 28
|
||||
2 * 2 * 7
|
||||
# group the primes
|
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|
|||
# get the product of all values
|
||||
6</code></pre>
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<p>Honestly, I have no idea why this gives as the number of divisors. I will implement the solution and then try to come up with an explanation.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_prime</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">smaller_primes</span><span class="p">):</span>
|
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<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">smaller_primes</span><span class="p">:</span>
|
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<span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="n">s</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
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<span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">remainder</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">factors</span>
|
||||
</pre></div>
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<p>This are the prime numbers related functions we already now. Now we implement a group function, the product function we already know, and based on that the algorithm to get the number of divisors mentioned above.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">group</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
|
||||
<span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">reduce</span>
|
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<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">xss</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">get_number_of_divisors</span><span class="p">(</span><span class="mi">28</span><span class="p">)</span> <span class="o">==</span> <span class="mi">6</span><span class="p">)</span>
|
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</pre></div>
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<p>Now we are ready to do another brute force attempt.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">ts</span> <span class="o">=</span> <span class="n">triangle_number_generator_function</span><span class="p">()</span>
|
||||
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">:</span>
|
||||
<span class="k">if</span> <span class="n">get_number_of_divisors</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">></span> <span class="mi">500</span><span class="p">:</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
|
||||
<span class="k">break</span>
|
||||
</pre></div>
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<pre>76576500
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<p>Now the only question is why this crazy algorithm works. Okay, I got it with the help of <a href="https://www.math.upenn.edu/~deturck/m170/wk2/numdivisors.html">this</a> page. The problem is actually an instance of the multiplication principle for counting things. Each prime can be used to calculate (n + 1) other divisors. The incrementation by one is required because we can also choose to not use a certain prime. Of course, to get the potential combinations we have to get the product of the potential divisors for all primes. The web page explains it better. The only question is whether I came up with this algorithm myself back then.</p>
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<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem013</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
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<style type="text/css">
|
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/*!
|
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*
|
||||
|
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|
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.ansi-bold { font-weight: bold; }
|
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|
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|
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|
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|
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|
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|
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<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
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<link rel="stylesheet" href="custom.css">
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||||
|
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<link href="custom.css" rel="stylesheet"/>
|
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||||
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|
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|
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|
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<script type="text/x-mathjax-config">
|
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|
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|
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|
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|
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||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
<p>Work out the first ten digits of the sum of the following one-hundred 50-digit numbers.</p>
|
||||
|
||||
<h1 id="Euler-Problem-13">Euler Problem 13<a class="anchor-link" href="#Euler-Problem-13">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>Work out the first ten digits of the sum of the following one-hundred 50-digit numbers.</p>
|
||||
<pre><code>37107287533902102798797998220837590246510135740250
|
||||
46376937677490009712648124896970078050417018260538
|
||||
74324986199524741059474233309513058123726617309629
|
||||
|
|
@ -11877,7 +11871,6 @@ div#notebook {
|
|||
72107838435069186155435662884062257473692284509516
|
||||
20849603980134001723930671666823555245252804609722
|
||||
53503534226472524250874054075591789781264330331690</code></pre>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
|
|
@ -11886,16 +11879,15 @@ div#notebook {
|
|||
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|
||||
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|
||||
<p>Straight forward. No big deal.</p>
|
||||
|
||||
</div>
|
||||
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|
||||
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|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">numbers_string</span> <span class="o">=</span> <span class="s2">"""</span>
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">numbers_string</span> <span class="o">=</span> <span class="s2">"""</span>
|
||||
<span class="s2">37107287533902102798797998220837590246510135740250</span>
|
||||
<span class="s2">46376937677490009712648124896970078050417018260538</span>
|
||||
<span class="s2">74324986199524741059474233309513058123726617309629</span>
|
||||
|
|
@ -11996,43 +11988,30 @@ div#notebook {
|
|||
<span class="s2">72107838435069186155435662884062257473692284509516</span>
|
||||
<span class="s2">20849603980134001723930671666823555245252804609722</span>
|
||||
<span class="s2">53503534226472524250874054075591789781264330331690</span>
|
||||
<span class="s2">"""</span>
|
||||
<span class="s2">"""</span>
|
||||
|
||||
<span class="n">numbers</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">numbers_string</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s1">'</span><span class="se">\n</span><span class="s1">'</span><span class="p">)</span> <span class="k">if</span> <span class="n">n</span><span class="p">]</span>
|
||||
<span class="n">numbers</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">numbers_string</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s1">'</span><span class="se">\n</span><span class="s1">'</span><span class="p">)</span> <span class="k">if</span> <span class="n">n</span><span class="p">]</span>
|
||||
<span class="n">numbers_sum</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">numbers</span><span class="p">)</span>
|
||||
<span class="n">first_ten_digits</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">numbers_sum</span><span class="p">)[:</span><span class="mi">10</span><span class="p">]</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">first_ten_digits</span> <span class="o">==</span> <span class="s2">"5537376230"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">first_ten_digits</span> <span class="o">==</span> <span class="s2">"5537376230"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">first_ten_digits</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
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|
||||
|
||||
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|
||||
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|
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|
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|
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|
||||
|
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|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>5537376230
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem014</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
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|
||||
|
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|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
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@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,16 +11762,15 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-14">Euler Problem 14<a class="anchor-link" href="#Euler-Problem-14">¶</a></h1><p>The following iterative sequence is defined for the set of positive integers:</p>
|
||||
<h1 id="Euler-Problem-14">Euler Problem 14<a class="anchor-link" href="#Euler-Problem-14">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The following iterative sequence is defined for the set of positive integers:</p>
|
||||
<p>n → n/2 (n is even)
|
||||
n → 3n + 1 (n is odd)</p>
|
||||
<p>Using the rule above and starting with 13, we generate the following sequence:</p>
|
||||
|
|
@ -11783,7 +11778,6 @@ n → 3n + 1 (n is odd)</p>
|
|||
<p>It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.</p>
|
||||
<p>Which starting number, under one million, produces the longest chain?</p>
|
||||
<p>NOTE: Once the chain starts the terms are allowed to go above one million.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11792,15 +11786,14 @@ n → 3n + 1 (n is odd)</p>
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>I would go for a recursive function with a cache here. This should give us good performance with acceptable memory footprint.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">lru_cache</span>
|
||||
<span class="kn">from</span> <span class="nn">collections</span> <span class="k">import</span> <span class="n">deque</span>
|
||||
|
||||
|
|
@ -11814,62 +11807,50 @@ n → 3n + 1 (n is odd)</p>
|
|||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">get_collatz_sequence</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span> <span class="o">==</span> <span class="p">[</span><span class="mi">13</span><span class="p">,</span> <span class="mi">40</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Isn't this a beautiful solution? We need a deque to appendleft. Of course, we could also use a regular list and reverse the result, but it is nicer to get the correct solution out right away. Now we simply force the solution.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">try</span><span class="p">:</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([(</span><span class="nb">len</span><span class="p">(</span><span class="n">get_collatz_sequence</span><span class="p">(</span><span class="n">i</span><span class="p">)),</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000000</span><span class="p">)])</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">837799</span><span class="p">)</span>
|
||||
<span class="k">except</span> <span class="n">RecursionError</span><span class="p">:</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Okay, we need a loop."</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Okay, we need a loop."</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>Okay, we need a loop.
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">cache</span> <span class="o">=</span> <span class="p">{}</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">get_collatz_sequence</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
|
|
@ -11900,45 +11881,31 @@ n → 3n + 1 (n is odd)</p>
|
|||
<span class="nb">print</span><span class="p">(</span><span class="n">timeit</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">brute_force</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="mi">10</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">brute_force</span><span class="p">())</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>11.4885111964837
|
||||
837799
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Here we go. Takes some time. We could optimize by caching only the length and not the complete list.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem015</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,19 +11762,17 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-15">Euler Problem 15<a class="anchor-link" href="#Euler-Problem-15">¶</a></h1><p>Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.</p>
|
||||
<p><img src="https://projecteuler.net/project/images/p015.gif" alt="problem picture"></p>
|
||||
<h1 id="Euler-Problem-15">Euler Problem 15<a class="anchor-link" href="#Euler-Problem-15">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.</p>
|
||||
<p><img alt="problem picture" src="https://projecteuler.net/project/images/p015.gif"/></p>
|
||||
<p>How many such routes are there through a 20×20 grid?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11788,21 +11782,19 @@ div#notebook {
|
|||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We should be able to get this with some thinking. Every move for a $nxn$ grid has to contain n downward and n rightward moves. So in total there are 2n moves. Now we only have to get all permutations available with 2n moves. Normally, we would calculate the factorial, but this only works if all symbols are different. In this case we always only have two symbols for the 2n moves.
|
||||
For each solution there exist a certain number of ways we can create this solution. Let's assume we have two right arrows called (1, 2) and two down arrows called (3, 4). We now have the following options to generate the solution in the upper left corner.</p>
|
||||
|
||||
<pre><code>1 2 3 4
|
||||
1 2 4 3
|
||||
2 1 3 4
|
||||
2 1 4 2</code></pre>
|
||||
<p>So this means if we calculate the number of ways using the factorial $(2\times2)! = 4! = 24$, four of the solutions are equal which gives us $\frac{24}{4} = 6$. The way to calculate the four should be $2! \times 2! = 4$, so our formula is $\frac{(2\times n)!}{n! \times n!}$. So let's try that.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_number_of_routes</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">factorial</span>
|
||||
<span class="k">return</span> <span class="n">factorial</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span><span class="p">)</span> <span class="o">//</span> <span class="p">(</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
|
||||
|
|
@ -11811,29 +11803,20 @@ For each solution there exist a certain number of ways we can create this soluti
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">get_number_of_routes</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span> <span class="o">==</span> <span class="mi">137846528820</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">get_number_of_routes</span><span class="p">(</span><span class="mi">20</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
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|
||||
|
||||
|
||||
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|
||||
|
||||
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|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>137846528820
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
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|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
|
|
@ -11843,15 +11826,10 @@ For each solution there exist a certain number of ways we can create this soluti
|
|||
<p>So the final formula is</p>
|
||||
<p>$n_{routes} = \frac{(d \times n)!}{(n!)^d}$</p>
|
||||
<p>where d is the number of dimensions and n is the size of the grid.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
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|
||||
</div>
|
||||
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|
||||
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||||
|
||||
|
||||
|
||||
|
||||
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|
||||
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|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
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||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem016</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
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|
||||
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||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
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@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,18 +11762,16 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
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||||
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||||
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|
||||
<h1 id="Euler-Problem-16">Euler Problem 16<a class="anchor-link" href="#Euler-Problem-16">¶</a></h1><p>$2^{15}$ = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.</p>
|
||||
<h1 id="Euler-Problem-16">Euler Problem 16<a class="anchor-link" href="#Euler-Problem-16">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>$2^{15}$ = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.</p>
|
||||
<p>What is the sum of the digits of the number $2^{1000}$?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11786,57 +11780,46 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Seriously? Okay, probably more difficult in C.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [17]:</div>
|
||||
<div class="prompt input_prompt">In [17]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="nb">str</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">1000</span><span class="p">)))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">1366</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
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|
||||
|
||||
|
||||
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|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>1366
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, let's do it without big ints.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [18]:</div>
|
||||
<div class="prompt input_prompt">In [18]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">double_number_string</span><span class="p">(</span><span class="n">number_string</span><span class="p">):</span>
|
||||
<span class="n">number_string</span> <span class="o">=</span> <span class="n">number_string</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="c1"># reverse</span>
|
||||
<span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
|
|
@ -11848,77 +11831,61 @@ div#notebook {
|
|||
<span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">next_digit</span><span class="p">))</span>
|
||||
<span class="k">if</span> <span class="n">carriage</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">carriage</span><span class="p">))</span>
|
||||
<span class="n">result</span> <span class="o">=</span> <span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">result</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||||
<span class="n">result</span> <span class="o">=</span> <span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">result</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||||
<span class="k">return</span> <span class="n">result</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">double_number_string</span><span class="p">(</span><span class="s2">"132"</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"264"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">double_number_string</span><span class="p">(</span><span class="s2">"965"</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"1930"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">double_number_string</span><span class="p">(</span><span class="s2">"132"</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"264"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">double_number_string</span><span class="p">(</span><span class="s2">"965"</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"1930"</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [19]:</div>
|
||||
<div class="prompt input_prompt">In [19]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">powers_of_two</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="sd">""" Retuns nth power of two as a string. """</span>
|
||||
<span class="sd">""" Retuns nth power of two as a string. """</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">n</span> <span class="o">>=</span> <span class="mi">0</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="s2">"1"</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="s2">"2"</span>
|
||||
<span class="k">return</span> <span class="s2">"1"</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="s2">"2"</span>
|
||||
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="n">double_number_string</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">s</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">powers_of_two</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"8"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">powers_of_two</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"1024"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">powers_of_two</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"8"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">powers_of_two</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"1024"</span><span class="p">)</span>
|
||||
|
||||
<span class="n">number_string</span> <span class="o">=</span> <span class="n">powers_of_two</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">number_string</span><span class="p">)))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>1366
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Here we go. The conversion to integer and the sum would be easy in C.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem017</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,19 +11762,17 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-17">Euler Problem 17<a class="anchor-link" href="#Euler-Problem-17">¶</a></h1><p>If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.</p>
|
||||
<h1 id="Euler-Problem-17">Euler Problem 17<a class="anchor-link" href="#Euler-Problem-17">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.</p>
|
||||
<p>If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words, how many letters would be used?</p>
|
||||
<p>NOTE: Do not count spaces or hyphens. For example, 342 (three hundred and forty-two) contains 23 letters and 115 (one hundred and fifteen) contains 20 letters. The use of "and" when writing out numbers is in compliance with British usage.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11787,50 +11781,49 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>I can see us doing a semi-automated approach here or we write a nice function. We probably write a nice function because we are geeks.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">single_digit_integer_to_spoken_language</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="s2">""</span>
|
||||
<span class="k">return</span> <span class="s2">""</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">n</span> <span class="o">></span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">n</span> <span class="o"><</span> <span class="mi">10</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="p">{</span><span class="mi">1</span><span class="p">:</span> <span class="s1">'one'</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span> <span class="s1">'two'</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span> <span class="s1">'three'</span><span class="p">,</span> <span class="mi">4</span><span class="p">:</span> <span class="s1">'four'</span><span class="p">,</span> <span class="mi">5</span><span class="p">:</span> <span class="s1">'five'</span><span class="p">,</span>
|
||||
<span class="mi">6</span><span class="p">:</span> <span class="s1">'six'</span><span class="p">,</span> <span class="mi">7</span><span class="p">:</span> <span class="s1">'seven'</span><span class="p">,</span> <span class="mi">8</span><span class="p">:</span> <span class="s1">'eight'</span><span class="p">,</span> <span class="mi">9</span><span class="p">:</span> <span class="s1">'nine'</span><span class="p">}[</span><span class="n">n</span><span class="p">]</span>
|
||||
<span class="k">return</span> <span class="p">{</span><span class="mi">1</span><span class="p">:</span> <span class="s1">'one'</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span> <span class="s1">'two'</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span> <span class="s1">'three'</span><span class="p">,</span> <span class="mi">4</span><span class="p">:</span> <span class="s1">'four'</span><span class="p">,</span> <span class="mi">5</span><span class="p">:</span> <span class="s1">'five'</span><span class="p">,</span>
|
||||
<span class="mi">6</span><span class="p">:</span> <span class="s1">'six'</span><span class="p">,</span> <span class="mi">7</span><span class="p">:</span> <span class="s1">'seven'</span><span class="p">,</span> <span class="mi">8</span><span class="p">:</span> <span class="s1">'eight'</span><span class="p">,</span> <span class="mi">9</span><span class="p">:</span> <span class="s1">'nine'</span><span class="p">}[</span><span class="n">n</span><span class="p">]</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">double_digit_integer_to_spoken_language</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">n</span> <span class="o">></span> <span class="mi">9</span> <span class="ow">and</span> <span class="n">n</span> <span class="o"><</span> <span class="mi">100</span><span class="p">)</span>
|
||||
<span class="k">try</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="p">{</span>
|
||||
<span class="mi">10</span><span class="p">:</span> <span class="s1">'ten'</span><span class="p">,</span> <span class="mi">11</span><span class="p">:</span> <span class="s1">'eleven'</span><span class="p">,</span> <span class="mi">12</span><span class="p">:</span> <span class="s1">'twelve'</span><span class="p">,</span> <span class="mi">13</span><span class="p">:</span> <span class="s1">'thirteen'</span><span class="p">,</span>
|
||||
<span class="mi">14</span><span class="p">:</span> <span class="s1">'fourteen'</span><span class="p">,</span> <span class="mi">15</span><span class="p">:</span> <span class="s1">'fifteen'</span><span class="p">,</span> <span class="mi">16</span><span class="p">:</span> <span class="s1">'sixteen'</span><span class="p">,</span>
|
||||
<span class="mi">17</span><span class="p">:</span> <span class="s1">'seventeen'</span><span class="p">,</span> <span class="mi">18</span><span class="p">:</span> <span class="s1">'eighteen'</span><span class="p">,</span> <span class="mi">19</span><span class="p">:</span> <span class="s1">'nineteen'</span><span class="p">}[</span><span class="n">n</span><span class="p">]</span>
|
||||
<span class="mi">10</span><span class="p">:</span> <span class="s1">'ten'</span><span class="p">,</span> <span class="mi">11</span><span class="p">:</span> <span class="s1">'eleven'</span><span class="p">,</span> <span class="mi">12</span><span class="p">:</span> <span class="s1">'twelve'</span><span class="p">,</span> <span class="mi">13</span><span class="p">:</span> <span class="s1">'thirteen'</span><span class="p">,</span>
|
||||
<span class="mi">14</span><span class="p">:</span> <span class="s1">'fourteen'</span><span class="p">,</span> <span class="mi">15</span><span class="p">:</span> <span class="s1">'fifteen'</span><span class="p">,</span> <span class="mi">16</span><span class="p">:</span> <span class="s1">'sixteen'</span><span class="p">,</span>
|
||||
<span class="mi">17</span><span class="p">:</span> <span class="s1">'seventeen'</span><span class="p">,</span> <span class="mi">18</span><span class="p">:</span> <span class="s1">'eighteen'</span><span class="p">,</span> <span class="mi">19</span><span class="p">:</span> <span class="s1">'nineteen'</span><span class="p">}[</span><span class="n">n</span><span class="p">]</span>
|
||||
<span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
|
||||
<span class="k">pass</span>
|
||||
<span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="p">{</span><span class="mi">2</span><span class="p">:</span> <span class="s1">'twenty'</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span> <span class="s1">'thirty'</span><span class="p">,</span> <span class="mi">4</span><span class="p">:</span> <span class="s1">'forty'</span><span class="p">,</span> <span class="mi">5</span><span class="p">:</span> <span class="s1">'fifty'</span><span class="p">,</span>
|
||||
<span class="mi">6</span><span class="p">:</span> <span class="s1">'sixty'</span><span class="p">,</span> <span class="mi">7</span><span class="p">:</span> <span class="s1">'seventy'</span><span class="p">,</span> <span class="mi">8</span><span class="p">:</span> <span class="s1">'eighty'</span><span class="p">,</span> <span class="mi">9</span><span class="p">:</span> <span class="s1">'ninety'</span><span class="p">}[</span><span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">)]</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="p">{</span><span class="mi">2</span><span class="p">:</span> <span class="s1">'twenty'</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span> <span class="s1">'thirty'</span><span class="p">,</span> <span class="mi">4</span><span class="p">:</span> <span class="s1">'forty'</span><span class="p">,</span> <span class="mi">5</span><span class="p">:</span> <span class="s1">'fifty'</span><span class="p">,</span>
|
||||
<span class="mi">6</span><span class="p">:</span> <span class="s1">'sixty'</span><span class="p">,</span> <span class="mi">7</span><span class="p">:</span> <span class="s1">'seventy'</span><span class="p">,</span> <span class="mi">8</span><span class="p">:</span> <span class="s1">'eighty'</span><span class="p">,</span> <span class="mi">9</span><span class="p">:</span> <span class="s1">'ninety'</span><span class="p">}[</span><span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">)]</span>
|
||||
<span class="n">b</span> <span class="o">=</span> <span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
|
||||
<span class="k">return</span> <span class="n">a</span> <span class="o">+</span> <span class="s1">'-'</span> <span class="o">+</span> <span class="n">b</span>
|
||||
<span class="k">return</span> <span class="n">a</span> <span class="o">+</span> <span class="s1">'-'</span> <span class="o">+</span> <span class="n">b</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">triple_digit_integer_to_spoken_language</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">1</span><span class="p">:]</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">single_digit_integer_to_spoken_language</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
|
||||
<span class="n">b</span> <span class="o">=</span> <span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">a</span> <span class="o">+</span> <span class="s2">" hundred"</span>
|
||||
<span class="k">return</span> <span class="n">a</span> <span class="o">+</span> <span class="s2">" hundred and "</span> <span class="o">+</span> <span class="n">b</span>
|
||||
<span class="k">return</span> <span class="n">a</span> <span class="o">+</span> <span class="s2">" hundred"</span>
|
||||
<span class="k">return</span> <span class="n">a</span> <span class="o">+</span> <span class="s2">" hundred and "</span> <span class="o">+</span> <span class="n">b</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">four_digit_integer_to_spoken_language</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">1</span><span class="p">:]</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="n">single_digit_integer_to_spoken_language</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
|
||||
<span class="n">b</span> <span class="o">=</span> <span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
|
||||
<span class="k">return</span> <span class="n">a</span> <span class="o">+</span> <span class="s2">" thousand "</span> <span class="o">+</span> <span class="n">b</span>
|
||||
<span class="k">return</span> <span class="n">a</span> <span class="o">+</span> <span class="s2">" thousand "</span> <span class="o">+</span> <span class="n">b</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">integer_to_spoken_language</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="n">l</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
|
||||
|
|
@ -11843,77 +11836,60 @@ div#notebook {
|
|||
<span class="k">elif</span> <span class="n">l</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">four_digit_integer_to_spoken_language</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
<span class="k">else</span><span class="p">:</span>
|
||||
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s2">"Length not supported."</span><span class="p">)</span>
|
||||
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s2">"Length not supported."</span><span class="p">)</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'five'</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">19</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'nineteen'</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">21</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'twenty-one'</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">210</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'two hundred and ten'</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">3000</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'three thousand '</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">8333</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'eight thousand three hundred and thirty-three'</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'five'</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">19</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'nineteen'</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">21</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'twenty-one'</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">210</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'two hundred and ten'</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">3000</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'three thousand '</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="mi">8333</span><span class="p">)</span> <span class="o">==</span> <span class="s1">'eight thousand three hundred and thirty-three'</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, I won't win a code golf contest but at least we can get the solution now.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">l</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1001</span><span class="p">)])</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s2">" "</span><span class="p">,</span> <span class="s2">""</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s2">"-"</span><span class="p">,</span> <span class="s2">""</span><span class="p">))</span>
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">l</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">integer_to_spoken_language</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1001</span><span class="p">)])</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s2">" "</span><span class="p">,</span> <span class="s2">""</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s2">"-"</span><span class="p">,</span> <span class="s2">""</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">l</span> <span class="o">==</span> <span class="mi">21124</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>21124
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Made the classical fourty/forty error I have already done four years ago. Some things don't change. Also I still do not really like this problem. The Haskell solution is actually way easier to read this time.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
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<title>EulerProblem018</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
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<h1 id="Euler-Problem-18">Euler Problem 18<a class="anchor-link" href="#Euler-Problem-18">¶</a></h1><p>By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.</p>
|
||||
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||||
<h1 id="Euler-Problem-18">Euler Problem 18<a class="anchor-link" href="#Euler-Problem-18">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.</p>
|
||||
<pre><code> 3
|
||||
7 4
|
||||
2 4 6
|
||||
8 5 9 3</code></pre>
|
||||
<p>That is, 3 + 7 + 4 + 9 = 23.</p>
|
||||
<p>Find the maximum total from top to bottom of the triangle below:</p>
|
||||
|
||||
<pre><code>75
|
||||
95 64
|
||||
17 47 82
|
||||
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63 66 04 68 89 53 67 30 73 16 69 87 40 31
|
||||
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23</code></pre>
|
||||
<p>NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, <a href="https://projecteuler.net/problem=67">Problem 67</a>, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)</p>
|
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@ -11810,22 +11802,19 @@ div#notebook {
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<p>This is incredibly simple we simply from bottom to top choosing the higher value for each mini-tree.</p>
|
||||
<p>For example,</p>
|
||||
|
||||
<pre><code> 95 64
|
||||
17 47 82</code></pre>
|
||||
<p>will become:</p>
|
||||
|
||||
<pre><code> 142 146</code></pre>
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||||
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<div class="cell border-box-sizing code_cell rendered">
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">t</span> <span class="o">=</span> <span class="s2">"""</span>
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<div class="input_area">
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">t</span> <span class="o">=</span> <span class="s2">"""</span>
|
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<span class="s2">75</span>
|
||||
<span class="s2">95 64</span>
|
||||
<span class="s2">17 47 82</span>
|
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|
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@ -11841,110 +11830,89 @@ div#notebook {
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<span class="s2">91 71 52 38 17 14 91 43 58 50 27 29 48</span>
|
||||
<span class="s2">63 66 04 68 89 53 67 30 73 16 69 87 40 31</span>
|
||||
<span class="s2">04 62 98 27 23 09 70 98 73 93 38 53 60 04 23</span>
|
||||
<span class="s2">"""</span>
|
||||
<span class="s2">"""</span>
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</pre></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">reduce_rows</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">):</span>
|
||||
<span class="sd">""" xs is lower row and ys is upper row """</span>
|
||||
<span class="sd">""" xs is lower row and ys is upper row """</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">ys</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="p">[</span><span class="nb">max</span><span class="p">([</span><span class="n">xs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">ys</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">xs</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">ys</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ys</span><span class="p">))]</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">reduce_rows</span><span class="p">([</span><span class="mi">17</span><span class="p">,</span> <span class="mi">47</span><span class="p">,</span> <span class="mi">82</span><span class="p">],</span> <span class="p">[</span><span class="mi">95</span><span class="p">,</span> <span class="mi">64</span><span class="p">])</span> <span class="o">==</span> <span class="p">[</span><span class="mi">142</span><span class="p">,</span> <span class="mi">146</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
<p>Okay, now all we have to do is the parsing and then a simple fold.</p>
|
||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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||||
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">xss</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">xs</span><span class="o">.</span><span class="n">split</span><span class="p">()))</span> <span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="n">t</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span> <span class="k">if</span> <span class="n">xs</span><span class="p">]</span>
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">xss</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">xs</span><span class="o">.</span><span class="n">split</span><span class="p">()))</span> <span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="n">t</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span> <span class="k">if</span> <span class="n">xs</span><span class="p">]</span>
|
||||
<span class="n">xss</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
|
||||
<span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">reduce</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="n">reduce</span><span class="p">(</span><span class="n">reduce_rows</span><span class="p">,</span> <span class="n">xss</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">xss</span><span class="p">[</span><span class="mi">0</span><span class="p">])[</span><span class="mi">0</span><span class="p">]</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">1074</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, let's put this into a nice function an then solve <a href="EulerProblem067">problem 67</a> right away.</p>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">find_greatest_path_sum_in_triangle_string</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
|
||||
<span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">reduce</span>
|
||||
<span class="n">xss</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">xs</span><span class="o">.</span><span class="n">split</span><span class="p">()))</span> <span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="n">ts</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span> <span class="k">if</span> <span class="n">xs</span><span class="p">]</span>
|
||||
<span class="n">xss</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">xs</span><span class="o">.</span><span class="n">split</span><span class="p">()))</span> <span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="n">ts</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span> <span class="k">if</span> <span class="n">xs</span><span class="p">]</span>
|
||||
<span class="n">xss</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
|
||||
<span class="n">r</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">:</span> <span class="p">[</span><span class="nb">max</span><span class="p">([</span><span class="n">xs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">ys</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">xs</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">ys</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ys</span><span class="p">))]</span>
|
||||
<span class="k">return</span> <span class="n">reduce</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">xss</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">xss</span><span class="p">[</span><span class="mi">0</span><span class="p">])[</span><span class="mi">0</span><span class="p">]</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">find_greatest_path_sum_in_triangle_string</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
|
||||
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|
||||
|
||||
|
||||
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|
||||
<pre>1074
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
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|||
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||||
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|
||||
<title>EulerProblem019</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
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|
||||
/*!
|
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|
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@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
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|
||||
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|
||||
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|
||||
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@ -11741,15 +11739,13 @@ div#notebook {
|
|||
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|
||||
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|
||||
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|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
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@ -11766,16 +11762,15 @@ div#notebook {
|
|||
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|
||||
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|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
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|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
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|
||||
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||||
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||||
<h1 id="Euler-Problem-19">Euler Problem 19<a class="anchor-link" href="#Euler-Problem-19">¶</a></h1><p>You are given the following information, but you may prefer to do some research for yourself.</p>
|
||||
<h1 id="Euler-Problem-19">Euler Problem 19<a class="anchor-link" href="#Euler-Problem-19">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>You are given the following information, but you may prefer to do some research for yourself.</p>
|
||||
<ul>
|
||||
<li>1 Jan 1900 was a Monday.</li>
|
||||
<li>Thirty days has September,
|
||||
|
|
@ -11787,7 +11782,6 @@ And on leap years, twenty-nine.</li>
|
|||
<li>A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400.</li>
|
||||
</ul>
|
||||
<p>How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11799,24 +11793,23 @@ And on leap years, twenty-nine.</li>
|
|||
<p>We have one hundred years which equal 1200 months. Every month starts with a weekday and $\frac{1}{7}$ of all weekdays are Sundays. So, 1200 divided by 7 is 171 point something.</p>
|
||||
<p>Which turns out to be the correct solution.</p>
|
||||
<p>Let's still code a solution because we were not that smart back then. We create two generators, zip, them and search for 1st and Sunday.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">weekday_generator_function</span><span class="p">():</span>
|
||||
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
|
||||
<span class="k">yield</span> <span class="s1">'Monday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Tuesday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Wednesday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Thursday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Friday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Saturday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Sunday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Monday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Tuesday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Wednesday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Thursday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Friday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Saturday'</span>
|
||||
<span class="k">yield</span> <span class="s1">'Sunday'</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">day_of_month_generator_function</span><span class="p">():</span>
|
||||
<span class="n">day</span><span class="p">,</span> <span class="n">month</span><span class="p">,</span> <span class="n">year</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1901</span>
|
||||
|
|
@ -11844,91 +11837,67 @@ And on leap years, twenty-nine.</li>
|
|||
|
||||
|
||||
<span class="n">ds</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span><span class="n">weekday_generator_function</span><span class="p">(),</span> <span class="n">day_of_month_generator_function</span><span class="p">())</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">len</span><span class="p">([</span><span class="mi">1</span> <span class="k">for</span> <span class="n">weekday</span><span class="p">,</span> <span class="n">date</span> <span class="ow">in</span> <span class="n">ds</span> <span class="k">if</span> <span class="n">weekday</span> <span class="o">==</span> <span class="s2">"Sunday"</span> <span class="ow">and</span> <span class="n">date</span> <span class="o">==</span> <span class="mi">1</span><span class="p">])</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">len</span><span class="p">([</span><span class="mi">1</span> <span class="k">for</span> <span class="n">weekday</span><span class="p">,</span> <span class="n">date</span> <span class="ow">in</span> <span class="n">ds</span> <span class="k">if</span> <span class="n">weekday</span> <span class="o">==</span> <span class="s2">"Sunday"</span> <span class="ow">and</span> <span class="n">date</span> <span class="o">==</span> <span class="mi">1</span><span class="p">])</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>172
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, this is actually kind of embarrassing. Why am I even zipping when I have the month right there in my function. Problem is, that my weekday generator starts with Monday, but Monday was the start in 1900 not in 1901. Let's try to do better. 1.1.1901 was a Tuesday, so we drop Monday from the generator.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">wds</span> <span class="o">=</span> <span class="n">weekday_generator_function</span><span class="p">()</span>
|
||||
<span class="nb">next</span><span class="p">(</span><span class="n">wds</span><span class="p">)</span> <span class="c1"># get rid of first Monday</span>
|
||||
<span class="n">ds</span> <span class="o">=</span> <span class="nb">zip</span><span class="p">(</span><span class="n">wds</span><span class="p">,</span> <span class="n">day_of_month_generator_function</span><span class="p">())</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">len</span><span class="p">([</span><span class="mi">1</span> <span class="k">for</span> <span class="n">weekday</span><span class="p">,</span> <span class="n">date</span> <span class="ow">in</span> <span class="n">ds</span> <span class="k">if</span> <span class="n">weekday</span> <span class="o">==</span> <span class="s2">"Sunday"</span> <span class="ow">and</span> <span class="n">date</span> <span class="o">==</span> <span class="mi">1</span><span class="p">])</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">len</span><span class="p">([</span><span class="mi">1</span> <span class="k">for</span> <span class="n">weekday</span><span class="p">,</span> <span class="n">date</span> <span class="ow">in</span> <span class="n">ds</span> <span class="k">if</span> <span class="n">weekday</span> <span class="o">==</span> <span class="s2">"Sunday"</span> <span class="ow">and</span> <span class="n">date</span> <span class="o">==</span> <span class="mi">1</span><span class="p">])</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>171
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>And finally, here we go. Actually not a dummy exercise and shows how imporessive the shortcut from the beginning is. Raw power is not always the best solution if we can also use brain.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem020</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,20 +11762,18 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-20">Euler Problem 20<a class="anchor-link" href="#Euler-Problem-20">¶</a></h1><p>n! means n × (n − 1) × ... × 3 × 2 × 1</p>
|
||||
<h1 id="Euler-Problem-20">Euler Problem 20<a class="anchor-link" href="#Euler-Problem-20">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>n! means n × (n − 1) × ... × 3 × 2 × 1</p>
|
||||
<p>For example, 10! = 10 × 9 × ... × 3 × 2 × 1 = 3628800,
|
||||
and the sum of the digits in the number 10! is 3 + 6 + 2 + 8 + 8 + 0 + 0 = 27.</p>
|
||||
<p>Find the sum of the digits in the number 100!</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11788,15 +11782,14 @@ and the sum of the digits in the number 10! is 3 + 6 + 2 + 8 + 8 + 0 + 0 = 27.</
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Write our own factorial implementation and get the solution.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">n</span> <span class="o">></span> <span class="mi">0</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
|
||||
|
|
@ -11804,50 +11797,35 @@ and the sum of the digits in the number 10! is 3 + 6 + 2 + 8 + 8 + 0 + 0 = 27.</
|
|||
<span class="k">else</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">n</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">f100</span> <span class="o">=</span> <span class="n">factorial</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="nb">str</span><span class="p">(</span><span class="n">f100</span><span class="p">))))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="nb">str</span><span class="p">(</span><span class="n">f100</span><span class="p">)))</span> <span class="o">==</span> <span class="mi">648</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>648
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem021</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,20 +11762,18 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-21">Euler Problem 21<a class="anchor-link" href="#Euler-Problem-21">¶</a></h1><p>Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
|
||||
<h1 id="Euler-Problem-21">Euler Problem 21<a class="anchor-link" href="#Euler-Problem-21">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
|
||||
If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are called amicable numbers.</p>
|
||||
<p>For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.</p>
|
||||
<p>Evaluate the sum of all the amicable numbers under 10000.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11788,15 +11782,14 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We reuse the sieve of eratosthenes (which we called get_primes_smaller previously). What annoys me about the implementation is that we do not stop scanning for prospects once the current prime squared is greater than the limit. Let's test if it returns the same result and compare the execution time.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_primes_smaller</span><span class="p">(</span><span class="n">limit</span><span class="p">):</span>
|
||||
<span class="n">primes</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">prospects</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">limit</span><span class="p">)]</span>
|
||||
|
|
@ -11824,45 +11817,35 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
<span class="nb">print</span><span class="p">(</span><span class="n">timeit</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="n">sieve_of_eratosthenes</span><span class="p">,</span> <span class="mi">10000</span><span class="p">),</span> <span class="n">number</span><span class="o">=</span><span class="mi">100</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_primes_smaller</span><span class="p">(</span><span class="mi">10000</span><span class="p">)</span> <span class="o">==</span> <span class="n">sieve_of_eratosthenes</span><span class="p">(</span><span class="mi">10000</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>5.062845234464119
|
||||
0.3188800166435408
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, this difference is sick actually. Interesting to see what the difference between O(sqrt(n)) and O(n) can mean. Now we can use the primes to do the factorization. I will again apply some optimizations and see how it turns out</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_prime_factors_old</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
|
||||
<span class="n">prime_generator</span> <span class="o">=</span> <span class="n">sieve_of_eratosthenes</span><span class="p">(</span><span class="n">number</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span>
|
||||
<span class="n">remainder</span> <span class="o">=</span> <span class="n">number</span>
|
||||
|
|
@ -11897,30 +11880,21 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
<span class="nb">print</span><span class="p">(</span><span class="n">timeit</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">functools</span><span class="o">.</span><span class="n">partial</span><span class="p">(</span><span class="n">get_prime_factors</span><span class="p">,</span> <span class="mi">1000000</span><span class="p">),</span> <span class="n">number</span><span class="o">=</span><span class="mi">10</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_prime_factors_old</span><span class="p">(</span><span class="mi">100000</span><span class="p">)</span> <span class="o">==</span> <span class="n">get_prime_factors</span><span class="p">(</span><span class="mi">100000</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>6.370271180404346
|
||||
5.785088868397899
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
|
|
@ -11929,15 +11903,14 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
<p>Okay, actually no big difference. Kind of expected if you check how similar the implementations are. I though stopping if p is greater than the remainder may be a common case.</p>
|
||||
<p>Now we can use the prime factors to get all factors. For this each prime number can be used 0 to n times where n is the number how often the prime occurs in the prime factorization. For example, all possible fators for the primes $2, 2, 3$ are $2^0 \times 3^0, 2^1 \times 3^0, 2^2 \times 3^0, 2^0 \times 3^1, 2^1 \times 3^1, 2^2 \times 3^1$ which results in $1, 2, 4, 3, 6, 12$.</p>
|
||||
<p>So we proceed in the following way. Once we have the prime factors we group then. Then we calculate the factors for each group. For example for $2,2$ the factors would be $1, 2, 4$ as explained in the previous paragraph. Then we calculate the combinations of all factor groups.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">group</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
|
||||
<span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">reduce</span>
|
||||
<span class="n">rs</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
|
|
@ -11962,11 +11935,9 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">group</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span> <span class="o">==</span> <span class="p">[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">]])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
|
|
@ -11974,15 +11945,14 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>As you can see I had a fancy version with reduce first, but the loop is way more readable. We iterate over the input list. If the current item is in the last sublist of the return list we add it to the list. Otherwise we append a new sublist with the item.</p>
|
||||
<p>Next we write a function that calculates the resulting factors for a factor group.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">prime_group_to_factors</span><span class="p">(</span><span class="n">group</span><span class="p">):</span>
|
||||
<span class="n">f</span> <span class="o">=</span> <span class="n">group</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">group</span><span class="p">)</span>
|
||||
|
|
@ -11990,68 +11960,60 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">prime_group_to_factors</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>No we only have to get all factors by calculating the combination of two factor lists.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [5]:</div>
|
||||
<div class="prompt input_prompt">In [5]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">combine_factors</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="p">[</span><span class="n">x</span> <span class="o">*</span> <span class="n">y</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">ys</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">]</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">combine_factors</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">12</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Now actually we want to combine an arbitrary number of factor lists.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [6]:</div>
|
||||
<div class="prompt input_prompt">In [6]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">combine_factors</span><span class="p">(</span><span class="n">xss</span><span class="p">):</span>
|
||||
<span class="n">ys</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||||
<span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="n">xss</span><span class="p">:</span>
|
||||
<span class="n">ys</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">*</span> <span class="n">y</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">ys</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">]</span>
|
||||
<span class="k">return</span> <span class="n">ys</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [7]:</div>
|
||||
<div class="prompt input_prompt">In [7]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_divisors</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="n">prime_factors</span> <span class="o">=</span> <span class="n">get_prime_factors</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
<span class="n">prime_groups</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="n">prime_factors</span><span class="p">)</span>
|
||||
|
|
@ -12064,26 +12026,23 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">get_divisors</span><span class="p">(</span><span class="mi">13</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">13</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_divisors</span><span class="p">(</span><span class="mi">26</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">26</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We can finally start to get the solution. Remember that we only want to get the sum of proper divisors which are the divisors withouth the number itself.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [8]:</div>
|
||||
<div class="prompt input_prompt">In [8]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">sum_of_proper_divisors</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">get_divisors</span><span class="p">(</span><span class="n">number</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
|
||||
|
||||
|
|
@ -12098,103 +12057,81 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_amicable</span><span class="p">(</span><span class="mi">220</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>The if statement is very import because a number pair $a, b$ is only amicable if $a \neq b$.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [9]:</div>
|
||||
<div class="prompt input_prompt">In [9]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">amicable_till_10000</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span> <span class="k">if</span> <span class="n">is_amicable</span><span class="p">(</span><span class="n">i</span><span class="p">)]</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">amicable_till_10000</span><span class="p">)</span> <span class="o">==</span> <span class="mi">31626</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">amicable_till_10000</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>31626
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>I found some brute force solutions in the Euler forum and they claim to have a sick performance. I want to compare them.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [10]:</div>
|
||||
<div class="prompt input_prompt">In [10]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">brute_force</span><span class="p">():</span>
|
||||
<span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span> <span class="k">if</span> <span class="n">is_amicable</span><span class="p">(</span><span class="n">i</span><span class="p">)])</span>
|
||||
|
||||
<span class="kn">import</span> <span class="nn">timeit</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">timeit</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">brute_force</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="mi">10</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>108.68903765424807
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
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||||
<div class="prompt input_prompt">In [14]:</div>
|
||||
<div class="prompt input_prompt">In [14]:</div>
|
||||
<div class="inner_cell">
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||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">sum_of_proper_divisors</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">sqrt</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
|
|
@ -12218,29 +12155,20 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
<span class="kn">import</span> <span class="nn">timeit</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">timeit</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">brute_force</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="mi">10</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>0.7862764837699387
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
|
|
@ -12248,15 +12176,10 @@ If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and e
|
|||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>This is actually really embarrassing. Especially I was aware of the sqrt trick all the way but did not manage to put it into use properly. Another example where greate engineer will leave you far behind compared to great thinking.</p>
|
||||
<p><strong>Supplement:</strong> Also I have discovered that my sum of proper divisors function has a bug when n is a square, because in that case <em>i == (n // 1)</em> which adds this value two times. So there is no amicable number under 10000 which is also a square because otherwise we would have failed.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem022</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,19 +11762,17 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-22">Euler Problem 22<a class="anchor-link" href="#Euler-Problem-22">¶</a></h1><p>Using names.txt (saved as EulerProblem022.txt in the same directory as this notebook), a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position in the list to obtain a name score.</p>
|
||||
<h1 id="Euler-Problem-22">Euler Problem 22<a class="anchor-link" href="#Euler-Problem-22">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>Using names.txt (saved as EulerProblem022.txt in the same directory as this notebook), a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position in the list to obtain a name score.</p>
|
||||
<p>For example, when the list is sorted into alphabetical order, COLIN, which is worth 3 + 15 + 12 + 9 + 14 = 53, is the 938th name in the list. So, COLIN would obtain a score of 938 × 53 = 49714.</p>
|
||||
<p>What is the total of all the name scores in the file?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11787,82 +11781,71 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, this should be straight forward.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [ ]:</div>
|
||||
<div class="prompt input_prompt">In [ ]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s1">'EulerProblem022.txt'</span><span class="p">,</span> <span class="s1">'r'</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
|
||||
<span class="n">names</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">read</span><span class="p">()</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s1">','</span><span class="p">)</span>
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s1">'EulerProblem022.txt'</span><span class="p">,</span> <span class="s1">'r'</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
|
||||
<span class="n">names</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">read</span><span class="p">()</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s1">','</span><span class="p">)</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">get_score_for_name</span><span class="p">(</span><span class="n">name</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="nb">ord</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">-</span> <span class="nb">ord</span><span class="p">(</span><span class="s1">'A'</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">name</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span> <span class="o">==</span> <span class="s1">'"'</span><span class="p">])</span>
|
||||
<span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="nb">ord</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">-</span> <span class="nb">ord</span><span class="p">(</span><span class="s1">'A'</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">name</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span> <span class="o">==</span> <span class="s1">'"'</span><span class="p">])</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_score_for_name</span><span class="p">(</span><span class="s1">'COLIN'</span><span class="p">)</span> <span class="o">==</span> <span class="mi">53</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_score_for_name</span><span class="p">(</span><span class="s1">'COLIN'</span><span class="p">)</span> <span class="o">==</span> <span class="mi">53</span><span class="p">)</span>
|
||||
<span class="n">names</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">get_score_for_name</span><span class="p">(</span><span class="n">name</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">name</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">names</span><span class="p">)])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">871198282</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>I think nothing to explain here. The only question is what was if we hadn't the Python sort function to sort into alphabetical order, then we would have to write our own compare function and use it with what ever sorting algorithm.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [19]:</div>
|
||||
<div class="prompt input_prompt">In [19]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">compare</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
|
||||
<span class="k">try</span><span class="p">:</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)):</span>
|
||||
<span class="k">if</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o"><</span> <span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
|
||||
<span class="k">return</span> <span class="s1">'</span><span class="si">{}</span><span class="s1"> before </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="s1">'</span><span class="si">{}</span><span class="s1"> before </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
|
||||
<span class="k">elif</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">></span> <span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
|
||||
<span class="k">return</span> <span class="s1">'</span><span class="si">{}</span><span class="s1"> before </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="s1">'</span><span class="si">{}</span><span class="s1"> before </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
|
||||
<span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
|
||||
<span class="k">pass</span>
|
||||
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o"><</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="s1">'</span><span class="si">{}</span><span class="s1"> before </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="s1">'</span><span class="si">{}</span><span class="s1"> before </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
|
||||
<span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">></span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="s1">'</span><span class="si">{}</span><span class="s1"> before </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="s1">'</span><span class="si">{}</span><span class="s1"> before </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
|
||||
<span class="k">else</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="s1">'</span><span class="si">{}</span><span class="s1"> is </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="s1">'</span><span class="si">{}</span><span class="s1"> is </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">compare</span><span class="p">(</span><span class="s1">'Felix'</span><span class="p">,</span> <span class="s1">'Arnold'</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">compare</span><span class="p">(</span><span class="s1">'Felix'</span><span class="p">,</span> <span class="s1">'Felixb'</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">compare</span><span class="p">(</span><span class="s1">'Felixb'</span><span class="p">,</span> <span class="s1">'Felix'</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">compare</span><span class="p">(</span><span class="s1">'Felix'</span><span class="p">,</span> <span class="s1">'Felix'</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">compare</span><span class="p">(</span><span class="s1">'Felix'</span><span class="p">,</span> <span class="s1">'Arnold'</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">compare</span><span class="p">(</span><span class="s1">'Felix'</span><span class="p">,</span> <span class="s1">'Felixb'</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">compare</span><span class="p">(</span><span class="s1">'Felixb'</span><span class="p">,</span> <span class="s1">'Felix'</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">compare</span><span class="p">(</span><span class="s1">'Felix'</span><span class="p">,</span> <span class="s1">'Felix'</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>Arnold before Felix
|
||||
Felix before Felixb
|
||||
|
|
@ -11871,25 +11854,18 @@ Felix is Felix
|
|||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Obviously, the algorithm would return True/False or 0/1/-1 for real sorting.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem023</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,20 +11762,18 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-23">Euler Problem 23<a class="anchor-link" href="#Euler-Problem-23">¶</a></h1><p>A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.</p>
|
||||
<h1 id="Euler-Problem-23">Euler Problem 23<a class="anchor-link" href="#Euler-Problem-23">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.</p>
|
||||
<p>A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.</p>
|
||||
<p>As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.</p>
|
||||
<p>Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11788,15 +11782,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We reuse the sum of proper divisors function and use it to tell whether a number is abundant or not.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [20]:</div>
|
||||
<div class="prompt input_prompt">In [20]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">sum_of_proper_divisors</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">sqrt</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
|
||||
|
|
@ -11819,26 +11812,23 @@ div#notebook {
|
|||
<span class="n">abundant_numbers_smaller_30000</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">30001</span><span class="p">)</span> <span class="k">if</span> <span class="n">is_abundant</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
|
||||
<span class="n">abundant_numbers_smaller_30000_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">abundant_numbers_smaller_30000</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Now that we have all abundant numbers in sorted order we can write a function which tests whether a number can be written as a sum of two abundant numbers.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [21]:</div>
|
||||
<div class="prompt input_prompt">In [21]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_sum_of_two_abundants</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">abundant_numbers_smaller_30000</span><span class="p">:</span>
|
||||
<span class="k">if</span> <span class="n">a</span> <span class="o">></span> <span class="p">(</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">):</span>
|
||||
|
|
@ -11852,59 +11842,43 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">is_sum_of_two_abundants</span><span class="p">(</span><span class="mi">23</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_sum_of_two_abundants</span><span class="p">(</span><span class="mi">28123</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Now we try to brute force.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [34]:</div>
|
||||
<div class="prompt input_prompt">In [34]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">30000</span><span class="p">)</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sum_of_two_abundants</span><span class="p">(</span><span class="n">i</span><span class="p">)])</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">4179871</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>4179871
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem024</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,19 +11762,17 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-24">Euler Problem 24<a class="anchor-link" href="#Euler-Problem-24">¶</a></h1><p>A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:</p>
|
||||
<h1 id="Euler-Problem-24">Euler Problem 24<a class="anchor-link" href="#Euler-Problem-24">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:</p>
|
||||
<p>012 021 102 120 201 210</p>
|
||||
<p>What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11787,15 +11781,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>I tried to solve this by thinking and failed to be hones. So we implement a generator and get the millionth element.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [19]:</div>
|
||||
<div class="prompt input_prompt">In [19]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">collections</span> <span class="k">import</span> <span class="n">deque</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">permutation_generator</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
|
||||
|
|
@ -11810,66 +11803,50 @@ div#notebook {
|
|||
<span class="k">yield</span> <span class="n">y</span>
|
||||
<span class="k">raise</span> <span class="ne">StopIteration</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">permutation_generator</span><span class="p">(</span><span class="s2">"012"</span><span class="p">)))</span> <span class="o">==</span> <span class="p">[</span><span class="s1">'012'</span><span class="p">,</span> <span class="s1">'021'</span><span class="p">,</span> <span class="s1">'102'</span><span class="p">,</span> <span class="s1">'120'</span><span class="p">,</span> <span class="s1">'201'</span><span class="p">,</span> <span class="s1">'210'</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">permutation_generator</span><span class="p">(</span><span class="s2">"012"</span><span class="p">)))</span> <span class="o">==</span> <span class="p">[</span><span class="s1">'012'</span><span class="p">,</span> <span class="s1">'021'</span><span class="p">,</span> <span class="s1">'102'</span><span class="p">,</span> <span class="s1">'120'</span><span class="p">,</span> <span class="s1">'201'</span><span class="p">,</span> <span class="s1">'210'</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>All right, now we reuse the function to get the nth element from problem 7 and give ourselves a solution.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [26]:</div>
|
||||
<div class="prompt input_prompt">In [26]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_nth</span><span class="p">(</span><span class="n">generator</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
|
||||
<span class="kn">import</span> <span class="nn">itertools</span>
|
||||
<span class="k">return</span> <span class="nb">next</span><span class="p">(</span><span class="n">itertools</span><span class="o">.</span><span class="n">islice</span><span class="p">(</span><span class="n">generator</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
|
||||
|
||||
<span class="n">ps</span> <span class="o">=</span> <span class="n">permutation_generator</span><span class="p">(</span><span class="s2">"0123456789"</span><span class="p">)</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">get_nth</span><span class="p">(</span><span class="n">ps</span><span class="p">,</span> <span class="mi">1000000</span><span class="p">)))</span>
|
||||
<span class="n">ps</span> <span class="o">=</span> <span class="n">permutation_generator</span><span class="p">(</span><span class="s2">"0123456789"</span><span class="p">)</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">get_nth</span><span class="p">(</span><span class="n">ps</span><span class="p">,</span> <span class="mi">1000000</span><span class="p">)))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">2783915460</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>2783915460
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem025</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,19 +11762,17 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-25">Euler Problem 25<a class="anchor-link" href="#Euler-Problem-25">¶</a></h1><p>The Fibonacci sequence is defined by the recurrence relation:</p>
|
||||
<h1 id="Euler-Problem-25">Euler Problem 25<a class="anchor-link" href="#Euler-Problem-25">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The Fibonacci sequence is defined by the recurrence relation:</p>
|
||||
<p>$F_n = F_{n−1} + F_{n−2}$, where $F_1 = 1$ and $F_2 = 1$.
|
||||
Hence the first 12 terms will be:</p>
|
||||
|
||||
<pre><code>F1 = 1
|
||||
F2 = 1
|
||||
F3 = 2
|
||||
|
|
@ -11793,7 +11787,6 @@ F11 = 89
|
|||
F12 = 144</code></pre>
|
||||
<p>The 12th term, $F_{12}$, is the first term to contain three digits.</p>
|
||||
<p>What is the index of the first term in the Fibonacci sequence to contain 1000 digits?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11802,15 +11795,14 @@ F12 = 144</code></pre>
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Reuse fibonacci generator and then straight forward.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">fibonacci_generator</span><span class="p">():</span>
|
||||
<span class="n">a</span> <span class="o">=</span> <span class="mi">0</span>
|
||||
<span class="n">b</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
|
|
@ -11827,35 +11819,22 @@ F12 = 144</code></pre>
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">4782</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>4782
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem026</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,17 +11762,15 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-26">Euler Problem 26<a class="anchor-link" href="#Euler-Problem-26">¶</a></h1><p>A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:</p>
|
||||
|
||||
<h1 id="Euler-Problem-26">Euler Problem 26<a class="anchor-link" href="#Euler-Problem-26">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:</p>
|
||||
<pre><code>1/2 = 0.5
|
||||
1/3 = 0.(3)
|
||||
1/4 = 0.25
|
||||
|
|
@ -11788,7 +11782,6 @@ div#notebook {
|
|||
1/10 = 0.1</code></pre>
|
||||
<p>Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle.</p>
|
||||
<p>Find the value of d < 1000 for which 1/d contains the longest recurring cycle in its decimal fraction part.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11798,15 +11791,14 @@ div#notebook {
|
|||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>The trick here is to identify a cycle. The easiest way I see to do this is to memorize the remainders. If we have a remainder that we occurred previously there is a cycle.</p>
|
||||
<p>Let's consider 1/3. The initial remainder is 10. For ten divided by three the new remainder is again ten (or one times ten). So we have a one-cycle of 3.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_cycle_count</span><span class="p">(</span><span class="n">nominator</span><span class="p">,</span> <span class="n">denominator</span><span class="p">):</span>
|
||||
<span class="kn">from</span> <span class="nn">itertools</span> <span class="k">import</span> <span class="n">count</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">nominator</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
|
||||
|
|
@ -11829,60 +11821,44 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">get_cycle_count</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_cycle_count</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>This is a simple divison algorithm. The only special thing is the remainder and that we remember when it occurs the first time. If a remainder occurrs for the second time we substract the position and thus have the lenght of the cycle. With this solution we should be efficient enough to brute force.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([(</span><span class="n">get_cycle_count</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="n">i</span><span class="p">)</span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)])</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">983</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>983
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem027</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,16 +11762,15 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-27">Euler Problem 27<a class="anchor-link" href="#Euler-Problem-27">¶</a></h1><p>Euler discovered the remarkable quadratic formula:</p>
|
||||
<h1 id="Euler-Problem-27">Euler Problem 27<a class="anchor-link" href="#Euler-Problem-27">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>Euler discovered the remarkable quadratic formula:</p>
|
||||
<p>$n^2 + n + 41$</p>
|
||||
<p>It turns out that the formula will produce 40 primes for the consecutive integer values 0≤n≤39. However, when $n=40$, $40^2 + 40 + 41 = 40(40+1)+41$ is divisible by $41$, and certainly when $n=41,41^2+41+41$ is clearly divisible by 41.</p>
|
||||
<p>The incredible formula $n^2−79n+1601$ was discovered, which produces 80 primes for the consecutive values 0≤n≤79. The product of the coefficients, $−79$ and $1601$, is $−126479$.</p>
|
||||
|
|
@ -11783,7 +11778,6 @@ div#notebook {
|
|||
<p>$n^2 + an +b$, where |a|<1000 and |b|≤1000</p>
|
||||
<p>where |n| is the modulus/absolute value of n e.g. |11|=11 and |−4|=4.</p>
|
||||
<p>Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n=0.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11804,15 +11798,14 @@ div#notebook {
|
|||
<p>$p^2 - p + 41 >= 1000$ True for $\forall p \in \mathbb{N}$</p>
|
||||
<p>So now we only have to check for the values p in range(-30, 32). Alternatively, for the example $p = 40$ was used, maybe the next smaller value $p = 31$ yields the correct solution:</p>
|
||||
<p>$s = a\times b = (-2 * 31 + 1) * (31^2 - 31 + 41) = -61 \times 971 = -59231$</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">lru_cache</span>
|
||||
|
||||
<span class="nd">@lru_cache</span><span class="p">(</span><span class="n">maxsize</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
|
||||
|
|
@ -11847,37 +11840,21 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="o">-</span><span class="mi">59231</span><span class="p">)</span>
|
||||
<span class="n">s</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt output_prompt">Out[1]:</div>
|
||||
|
||||
|
||||
|
||||
|
||||
<div class="output_text output_subarea output_execute_result">
|
||||
<pre>-59231</pre>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem028</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,17 +11762,15 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-28">Euler Problem 28<a class="anchor-link" href="#Euler-Problem-28">¶</a></h1><p>Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:</p>
|
||||
|
||||
<h1 id="Euler-Problem-28">Euler Problem 28<a class="anchor-link" href="#Euler-Problem-28">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:</p>
|
||||
<pre><code>21 22 23 24 25
|
||||
20 7 8 9 10
|
||||
19 6 1 2 11
|
||||
|
|
@ -11785,7 +11779,6 @@ div#notebook {
|
|||
<p>$1 + 3 + 5 + 7 + 9 + 13 + 17 + 21 + 25 = 101$</p>
|
||||
<p>It can be verified that the sum of the numbers on the diagonals is 101.</p>
|
||||
<p>What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11794,15 +11787,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>When we know the corner of a certain spiral we can calculate it's total like $f_n = 4 c_n + 6 (n - 1)$. We then only have to update the corner value for each spiral.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">total</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<span class="n">current_corner</span> <span class="o">=</span> <span class="mi">3</span>
|
||||
|
||||
|
|
@ -11812,45 +11804,31 @@ div#notebook {
|
|||
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="n">total</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">669171001</span><span class="p">)</span>
|
||||
<span class="n">s</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt output_prompt">Out[2]:</div>
|
||||
|
||||
|
||||
|
||||
|
||||
<div class="output_text output_subarea output_execute_result">
|
||||
<pre>669171001</pre>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
|
|
@ -11862,50 +11840,33 @@ div#notebook {
|
|||
<p>$c_n = (n - 1)^2 - (n - 2) = n^2 - 2n + 1 - n + 2 = n^2 - 3n + 3$.</p>
|
||||
<p>Now, we can insert $c_n$ into $f_n$ which gives as the sum of corners for the nth spiral:</p>
|
||||
<p>$f_n = 4n^2 - 12n + 12 + 6n - 6 = 4n^2 - 6n + 6$</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="nb">sum</span><span class="p">([</span><span class="mi">4</span> <span class="o">*</span> <span class="n">n</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">6</span> <span class="o">*</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">6</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1002</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">669171001</span><span class="p">)</span>
|
||||
<span class="n">s</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt output_prompt">Out[3]:</div>
|
||||
|
||||
|
||||
|
||||
|
||||
<div class="output_text output_subarea output_execute_result">
|
||||
<pre>669171001</pre>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem029</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,16 +11762,15 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
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|
||||
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|
||||
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|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-29">Euler Problem 29<a class="anchor-link" href="#Euler-Problem-29">¶</a></h1><p>Consider all integer combinations of ab for 2 ≤ a ≤ 5 and 2 ≤ b ≤ 5:</p>
|
||||
<h1 id="Euler-Problem-29">Euler Problem 29<a class="anchor-link" href="#Euler-Problem-29">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>Consider all integer combinations of ab for 2 ≤ a ≤ 5 and 2 ≤ b ≤ 5:</p>
|
||||
<p>$2^2=4, 2^3=8, 2^4=16, 2^5=32$</p>
|
||||
<p>$3^2=9, 3^3=27, 3^4=81, 3^5=243$</p>
|
||||
<p>$4^2=16, 4^3=64, 4^4=256, 4^5=1024$</p>
|
||||
|
|
@ -11783,48 +11778,34 @@ div#notebook {
|
|||
<p>If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:</p>
|
||||
<p>$4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125$</p>
|
||||
<p>How many distinct terms are in the sequence generated by $a^b$ for $2 ≤ a ≤ 100$ and $2 ≤ b ≤ 100$?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="n">a</span><span class="o">**</span><span class="n">b</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">101</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">101</span><span class="p">)]))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">9183</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>9183
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem030</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,23 +11762,21 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-30">Euler Problem 30<a class="anchor-link" href="#Euler-Problem-30">¶</a></h1><p>Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:</p>
|
||||
<h1 id="Euler-Problem-30">Euler Problem 30<a class="anchor-link" href="#Euler-Problem-30">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:</p>
|
||||
<p>$1634 = 1^4 + 6^4 + 3^4 + 4^4$</p>
|
||||
<p>$8208 = 8^4 + 2^4 + 0^4 + 8^4$</p>
|
||||
<p>$9474 = 9^4 + 4^4 + 7^4 + 4^4$</p>
|
||||
<p>As $1 = 1^4$ is not a sum it is not included.</p>
|
||||
<p>The sum of these numbers is $1634 + 8208 + 9474 = 19316$.</p>
|
||||
<p>Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11791,15 +11785,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>This is an eays brute force. We look up the fifth powers.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [7]:</div>
|
||||
<div class="prompt input_prompt">In [7]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fifth_power_lookup</span> <span class="o">=</span> <span class="p">{</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">):</span> <span class="n">i</span><span class="o">**</span><span class="mi">5</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">)}</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">is_number_sum_of_fiths_powers_of_digits</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
|
|
@ -11809,35 +11802,22 @@ div#notebook {
|
|||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">443839</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>443839
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem031</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,21 +11762,19 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-31">Euler Problem 31<a class="anchor-link" href="#Euler-Problem-31">¶</a></h1><p>In England the currency is made up of pound, £, and pence, p, and there are eight coins in general circulation:</p>
|
||||
<h1 id="Euler-Problem-31">Euler Problem 31<a class="anchor-link" href="#Euler-Problem-31">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>In England the currency is made up of pound, £, and pence, p, and there are eight coins in general circulation:</p>
|
||||
<p>1p, 2p, 5p, 10p, 20p, 50p, £1 (100p) and £2 (200p).</p>
|
||||
<p>It is possible to make £2 in the following way:</p>
|
||||
<p>1×£1 + 1×50p + 2×20p + 1×5p + 1×2p + 3×1p</p>
|
||||
<p>How many different ways can £2 be made using any number of coins?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11789,50 +11783,40 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Let's do it non-recursive. Tricky part is to use ceil, otherwise we get wrong ranges for example for $\frac{50}{20}$.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">ceil</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">50</span> <span class="o">//</span> <span class="mi">20</span><span class="p">)))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="mi">50</span> <span class="o">/</span> <span class="mi">20</span><span class="p">))))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>[0, 1]
|
||||
[0, 1, 2]
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">ceil</span>
|
||||
|
||||
<span class="n">r</span> <span class="o">=</span> <span class="mi">200</span>
|
||||
|
|
@ -11873,49 +11857,34 @@ div#notebook {
|
|||
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="n">c</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">73682</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>73682
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem032</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,20 +11762,18 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-32">Euler Problem 32<a class="anchor-link" href="#Euler-Problem-32">¶</a></h1><p>We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once; for example, the 5-digit number, 15234, is 1 through 5 pandigital.</p>
|
||||
<h1 id="Euler-Problem-32">Euler Problem 32<a class="anchor-link" href="#Euler-Problem-32">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once; for example, the 5-digit number, 15234, is 1 through 5 pandigital.</p>
|
||||
<p>The product 7254 is unusual, as the identity, 39 × 186 = 7254, containing multiplicand, multiplier, and product is 1 through 9 pandigital.</p>
|
||||
<p>Find the sum of all products whose multiplicand/multiplier/product identity can be written as a 1 through 9 pandigital.</p>
|
||||
<p>HINT: Some products can be obtained in more than one way so be sure to only include it once in your sum.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11788,15 +11782,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We start by thinking about how many digits are used for multiplicand/multiplier/product. Obviously, the product must have more then three digits. Can it have more then 4 digits? Obviously it can because then there are only four digits for multiplicand and multiplier which is always smaller than ten thousand. So our product must be in the form $12 \cdot 345 = 5789$. This makes the rest very easy in Python.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">itertools</span> <span class="k">import</span> <span class="n">permutations</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">is_solution</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
|
||||
|
|
@ -11808,45 +11801,29 @@ div#notebook {
|
|||
<span class="k">return</span> <span class="n">c</span>
|
||||
<span class="k">return</span> <span class="mi">0</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_solution</span><span class="p">(</span><span class="s2">"391867254"</span><span class="p">)</span> <span class="o">==</span> <span class="mi">7254</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_solution</span><span class="p">(</span><span class="s2">"391867245"</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_solution</span><span class="p">(</span><span class="s2">"391867254"</span><span class="p">)</span> <span class="o">==</span> <span class="mi">7254</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_solution</span><span class="p">(</span><span class="s2">"391867245"</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
|
||||
|
||||
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="n">is_solution</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">p</span><span class="p">))</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">permutations</span><span class="p">(</span><span class="s2">"123456789"</span><span class="p">)]))</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="n">is_solution</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">p</span><span class="p">))</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">permutations</span><span class="p">(</span><span class="s2">"123456789"</span><span class="p">)]))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">45228</span><span class="p">)</span>
|
||||
<span class="n">s</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt output_prompt">Out[1]:</div>
|
||||
|
||||
|
||||
|
||||
|
||||
<div class="output_text output_subarea output_execute_result">
|
||||
<pre>45228</pre>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem033</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,20 +11762,18 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-33">Euler Problem 33<a class="anchor-link" href="#Euler-Problem-33">¶</a></h1><p>The fraction 49/98 is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which is correct, is obtained by cancelling the 9s.</p>
|
||||
<h1 id="Euler-Problem-33">Euler Problem 33<a class="anchor-link" href="#Euler-Problem-33">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The fraction 49/98 is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which is correct, is obtained by cancelling the 9s.</p>
|
||||
<p>We shall consider fractions like, 30/50 = 3/5, to be trivial examples.</p>
|
||||
<p>There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.</p>
|
||||
<p>If the product of these four fractions is given in its lowest common terms, find the value of the denominator.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11788,15 +11782,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We start write a function which checks if a number is curios and then brute force.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [21]:</div>
|
||||
<div class="prompt input_prompt">In [21]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_curious</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">))</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">d</span><span class="p">))</span> <span class="o">==</span> <span class="mi">2</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">d</span><span class="p">:</span>
|
||||
|
|
@ -11804,8 +11797,8 @@ div#notebook {
|
|||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">str</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
|
||||
<span class="k">try</span><span class="p">:</span>
|
||||
<span class="n">n_</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="s2">""</span><span class="p">))</span>
|
||||
<span class="n">d_</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">d</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="s2">""</span><span class="p">))</span>
|
||||
<span class="n">n_</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="s2">""</span><span class="p">))</span>
|
||||
<span class="n">d_</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">d</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="s2">""</span><span class="p">))</span>
|
||||
<span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">try</span><span class="p">:</span>
|
||||
|
|
@ -11818,97 +11811,73 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">is_curious</span><span class="p">(</span><span class="mi">49</span><span class="p">,</span> <span class="mi">98</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_curious</span><span class="p">(</span><span class="mi">30</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [25]:</div>
|
||||
<div class="prompt input_prompt">In [25]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">fs</span> <span class="o">=</span> <span class="p">[(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span> <span class="k">if</span> <span class="n">is_curious</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">)]</span>
|
||||
<span class="n">fs</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt output_prompt">Out[25]:</div>
|
||||
|
||||
|
||||
|
||||
|
||||
<div class="output_text output_subarea output_execute_result">
|
||||
<pre>[(16, 64), (19, 95), (26, 65), (49, 98)]</pre>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [32]:</div>
|
||||
<div class="prompt input_prompt">In [32]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<span class="n">d</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<span class="k">for</span> <span class="n">n_</span><span class="p">,</span> <span class="n">d_</span> <span class="ow">in</span> <span class="n">fs</span><span class="p">:</span>
|
||||
<span class="n">n</span> <span class="o">*=</span> <span class="n">n_</span>
|
||||
<span class="n">d</span> <span class="o">*=</span> <span class="n">d_</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="si">{}</span><span class="s2">/</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="si">{}</span><span class="s2">/</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
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|
||||
|
||||
|
||||
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|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>387296/38729600
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
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|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Now we can see that the solution is $100$. But actually it would be nice to calculate the GCD.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [40]:</div>
|
||||
<div class="prompt input_prompt">In [40]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">gcd_euclid</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">a</span>
|
||||
|
|
@ -11923,37 +11892,21 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">100</span><span class="p">)</span>
|
||||
<span class="n">s</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt output_prompt">Out[40]:</div>
|
||||
|
||||
|
||||
|
||||
|
||||
<div class="output_text output_subarea output_execute_result">
|
||||
<pre>100</pre>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem034</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
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|
||||
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|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,19 +11762,17 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
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|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-34">Euler Problem 34<a class="anchor-link" href="#Euler-Problem-34">¶</a></h1><p>145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.</p>
|
||||
<h1 id="Euler-Problem-34">Euler Problem 34<a class="anchor-link" href="#Euler-Problem-34">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.</p>
|
||||
<p>Find the sum of all numbers which are equal to the sum of the factorial of their digits.</p>
|
||||
<p>Note: as 1! = 1 and 2! = 2 are not sums they are not included.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11787,15 +11781,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>The algorithm for checking if a number is curious should be efficient. The more difficult thing is to select the upper bound for the brute force. It can be seen that $9 999 999 < 9! * 7$. Hence we can select $10^7$ as our bound.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">factorial</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">is_curious</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
|
|
@ -11804,64 +11797,44 @@ div#notebook {
|
|||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_curious</span><span class="p">(</span><span class="mi">145</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">7</span><span class="p">)</span> <span class="k">if</span> <span class="n">is_curious</span><span class="p">(</span><span class="n">n</span><span class="p">)])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
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|
||||
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|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">40730</span><span class="p">)</span>
|
||||
<span class="n">s</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
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|
||||
|
||||
<div class="prompt output_prompt">Out[3]:</div>
|
||||
|
||||
|
||||
|
||||
|
||||
<div class="output_text output_subarea output_execute_result">
|
||||
<pre>40730</pre>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem035</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,19 +11762,17 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-35">Euler Problem 35<a class="anchor-link" href="#Euler-Problem-35">¶</a></h1><p>The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.</p>
|
||||
<h1 id="Euler-Problem-35">Euler Problem 35<a class="anchor-link" href="#Euler-Problem-35">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.</p>
|
||||
<p>There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.</p>
|
||||
<p>How many circular primes are there below one million?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11787,15 +11781,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>First we get all primes into a look-up table. Then we iterate them and check whether they are circular.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_primes_smaller</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
|
||||
<span class="n">primes</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">prospects</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">number</span><span class="p">)]</span>
|
||||
|
|
@ -11809,17 +11802,15 @@ div#notebook {
|
|||
|
||||
<span class="n">ps</span> <span class="o">=</span> <span class="n">get_primes_smaller</span><span class="p">(</span><span class="mi">1000000</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_combinations</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="n">xs</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="p">[]</span>
|
||||
|
|
@ -11832,85 +11823,71 @@ div#notebook {
|
|||
<span class="n">rs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">xs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">ys</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">rs</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_combinations</span><span class="p">(</span><span class="s2">"ab"</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="s2">"ab"</span><span class="p">,</span> <span class="s2">"ba"</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_combinations</span><span class="p">(</span><span class="s2">"ab"</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="s2">"ab"</span><span class="p">,</span> <span class="s2">"ba"</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">itertools</span> <span class="k">import</span> <span class="n">permutations</span>
|
||||
<span class="n">prime_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">ps</span><span class="p">)</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">is_circular</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
|
||||
<span class="n">cs</span> <span class="o">=</span> <span class="n">permutations</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">cs</span><span class="p">:</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="nb">int</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">c</span><span class="p">))</span> <span class="ow">in</span> <span class="n">prime_set</span><span class="p">:</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="nb">int</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">c</span><span class="p">))</span> <span class="ow">in</span> <span class="n">prime_set</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">return</span> <span class="kc">True</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_circular</span><span class="p">(</span><span class="s2">"2"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_circular</span><span class="p">(</span><span class="s2">"11"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_circular</span><span class="p">(</span><span class="s2">"47"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_circular</span><span class="p">(</span><span class="s2">"2"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_circular</span><span class="p">(</span><span class="s2">"11"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_circular</span><span class="p">(</span><span class="s2">"47"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="nb">len</span><span class="p">([</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">ps</span> <span class="k">if</span> <span class="n">is_circular</span><span class="p">(</span><span class="n">p</span><span class="p">)])</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"False solution </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"False solution </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>False solution 22
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We did not read the problem properly. Cycles are obviously not the same as permutations. If we change that we should get the solution in no time.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [5]:</div>
|
||||
<div class="prompt input_prompt">In [5]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">cyles</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span> <span class="o"><=</span> <span class="mi">1</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">xs</span>
|
||||
|
|
@ -11921,55 +11898,37 @@ div#notebook {
|
|||
<span class="k">def</span> <span class="nf">is_circular</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
|
||||
<span class="n">cs</span> <span class="o">=</span> <span class="n">cyles</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
|
||||
<span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">cs</span><span class="p">:</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="nb">int</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">c</span><span class="p">))</span> <span class="ow">in</span> <span class="n">prime_set</span><span class="p">:</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="nb">int</span><span class="p">(</span><span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">c</span><span class="p">))</span> <span class="ow">in</span> <span class="n">prime_set</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">return</span> <span class="kc">True</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [6]:</div>
|
||||
<div class="prompt input_prompt">In [6]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="nb">len</span><span class="p">([</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">ps</span> <span class="k">if</span> <span class="n">is_circular</span><span class="p">(</span><span class="n">p</span><span class="p">)])</span>
|
||||
<span class="n">s</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt output_prompt">Out[6]:</div>
|
||||
|
||||
|
||||
|
||||
|
||||
<div class="output_text output_subarea output_execute_result">
|
||||
<pre>55</pre>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem036</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,19 +11762,17 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-36">Euler Problem 36<a class="anchor-link" href="#Euler-Problem-36">¶</a></h1><p>The decimal number, 585 = $1001001001_2$ (binary), is palindromic in both bases.</p>
|
||||
<h1 id="Euler-Problem-36">Euler Problem 36<a class="anchor-link" href="#Euler-Problem-36">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The decimal number, 585 = $1001001001_2$ (binary), is palindromic in both bases.</p>
|
||||
<p>Find the sum of all numbers, less than one million, which are palindromic in base 10 and base 2.</p>
|
||||
<p>(Please note that the palindromic number, in either base, may not include leading zeros.)</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11787,15 +11781,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>This looks like an easy problem to me. Like really easy. Should we try to brute force first?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_palindrome_decimal</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||||
|
||||
|
|
@ -11803,17 +11796,15 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">is_palindrome_decimal</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_palindrome_decimal</span><span class="p">(</span><span class="mi">2322</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_palindrome_binary</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="nb">bin</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">2</span><span class="p">:])</span>
|
||||
<span class="k">return</span> <span class="n">s</span> <span class="o">==</span> <span class="n">s</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
|
||||
|
|
@ -11821,61 +11812,44 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">is_palindrome_binary</span><span class="p">(</span><span class="mi">585</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_palindrome_binary</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000000</span><span class="p">)</span> <span class="k">if</span> <span class="n">is_palindrome_decimal</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="ow">and</span> <span class="n">is_palindrome_binary</span><span class="p">(</span><span class="n">i</span><span class="p">)])</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>872187
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem037</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
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|
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|
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<h1 id="Euler-Problem-37">Euler Problem 37<a class="anchor-link" href="#Euler-Problem-37">¶</a></h1><p>The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.</p>
|
||||
<h1 id="Euler-Problem-37">Euler Problem 37<a class="anchor-link" href="#Euler-Problem-37">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.</p>
|
||||
<p>Find the sum of the only eleven primes that are both truncatable from left to right and right to left.</p>
|
||||
<p>NOTE: 2, 3, 5, and 7 are not considered to be truncatable primes.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
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|
||||
|
|
@ -11788,15 +11782,14 @@ div#notebook {
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|||
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<p>Okay, I would say we start with one digit and work ourselves up. I will start with implementing a function that takes all current solutions (with one digit for example) and then tests all possible new solutions (aka all two digit primes) if the can be truncated to the one digit solutions.</p>
|
||||
<p>This was complicated. Let's formulate it clearer. We write a function that takes a list of numbers and tests whether they are left/right trunctable to another list of numbers.</p>
|
||||
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||||
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|
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_truncatable_numbers</span><span class="p">(</span><span class="n">numbers</span><span class="p">,</span> <span class="n">targets</span><span class="p">):</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">targets</span><span class="p">)</span>
|
||||
<span class="n">r</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
|
|
@ -11809,26 +11802,23 @@ div#notebook {
|
|||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_truncatable_numbers</span><span class="p">([</span><span class="mi">37</span><span class="p">,</span> <span class="mi">77</span><span class="p">,</span> <span class="mi">83</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span> <span class="o">==</span> <span class="p">[</span><span class="mi">37</span><span class="p">,</span> <span class="mi">77</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
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||||
</div>
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||||
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||||
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||||
<p>The next function returns a list of all primes with a certain number of digits n. We gonna reuse the sieve from problem 21.</p>
|
||||
|
||||
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|
||||
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|
||||
</div>
|
||||
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||||
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||||
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">sieve_of_eratosthenes</span><span class="p">(</span><span class="n">limit</span><span class="p">):</span>
|
||||
<span class="n">primes</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">prospects</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">limit</span><span class="p">)]</span>
|
||||
|
|
@ -11852,26 +11842,23 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">get_primes_with_n_digits</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">get_primes_with_n_digits</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="o">==</span> <span class="mi">21</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<p>Now we simply start with all one digit primes and work our way up using the two functions we have created. On the way up we add the numbers to our result list. For example, 97 is already a valid solution even though it is also the subset of a solution with more digits.</p>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">digits</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<span class="n">solutions</span> <span class="o">=</span> <span class="n">get_primes_with_n_digits</span><span class="p">(</span><span class="n">digits</span><span class="p">)</span>
|
||||
<span class="n">results</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
|
|
@ -11884,44 +11871,34 @@ div#notebook {
|
|||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
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|
||||
|
||||
|
||||
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|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>[23, 37, 53, 73, 373]
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
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|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
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|
||||
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|
||||
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|
||||
<p>This are not the eleven numbers we are looking for. The mistake we made is to not differentiate between left truncatable numbers and right truncatable numbers. For example, 397 is truncatable from the left, but not from the right and is still part of a solution. What we want to do is to work up from both directions and then compute the subset. Hence we write to new functions to check for all truncatable numbers from either left or right and then apply the algorithm from above to both of them.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="inner_cell">
|
||||
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|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_truncatable_numbers_right</span><span class="p">(</span><span class="n">numbers</span><span class="p">,</span> <span class="n">targets</span><span class="p">):</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">targets</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">numbers</span> <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="ow">in</span> <span class="n">s</span><span class="p">]</span>
|
||||
|
|
@ -11930,26 +11907,23 @@ div#notebook {
|
|||
<span class="n">s</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">targets</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">numbers</span> <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">1</span><span class="p">:])</span> <span class="ow">in</span> <span class="n">s</span><span class="p">]</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Let's redo the alogirthm.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [5]:</div>
|
||||
<div class="prompt input_prompt">In [5]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">digits</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<span class="n">solutions</span> <span class="o">=</span> <span class="n">get_primes_with_n_digits</span><span class="p">(</span><span class="n">digits</span><span class="p">)</span>
|
||||
<span class="n">results_left</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
|
|
@ -11976,45 +11950,35 @@ div#notebook {
|
|||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">results_right</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>[]
|
||||
[]
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
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|
||||
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|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>I added the terminate symbols because this algorithm did not even terminate. This means there are fairly big primes in either direction. We change the approach from bottom up into the other direction. Let's get all primes till a certain value and just check for them. Aka brute force. For this purpose we write a function that takes a number and checks if it is truncatable in both directions.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [6]:</div>
|
||||
<div class="prompt input_prompt">In [6]:</div>
|
||||
<div class="inner_cell">
|
||||
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|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">primes_till_10000</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">sieve_of_eratosthenes</span><span class="p">(</span><span class="mi">10000</span><span class="p">))</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">is_right_truncatable</span><span class="p">(</span><span class="n">number</span><span class="p">,</span> <span class="n">targets</span><span class="p">):</span>
|
||||
|
|
@ -12048,67 +12012,54 @@ div#notebook {
|
|||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_truncatable</span><span class="p">(</span><span class="mi">3797</span><span class="p">,</span> <span class="n">primes_till_10000</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Let's just go for a brute force till 10k and see what we get.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [7]:</div>
|
||||
<div class="prompt input_prompt">In [7]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">primes_till_10000</span> <span class="k">if</span> <span class="n">is_truncatable</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">primes_till_10000</span><span class="p">)]</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>[2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797]
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>The one digit numbers are not relevant but this means we got only ten solutions. So we actually have to try all numbers till $10^{6}$. As it turned out we actually have to add one more digit. This is really ugly brute force. I do not like it. But seems like we got a solution.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [8]:</div>
|
||||
<div class="prompt input_prompt">In [8]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">primes_till_100000</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">sieve_of_eratosthenes</span><span class="p">(</span><span class="mi">1000000</span><span class="p">))</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">primes_till_100000</span> <span class="k">if</span> <span class="n">is_truncatable</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">primes_till_100000</span><span class="p">)]</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
|
|
@ -12116,36 +12067,23 @@ div#notebook {
|
|||
<span class="n">s</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">4</span><span class="p">:])</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">748317</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
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|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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||||
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||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>[2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397]
|
||||
748317
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem038</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
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|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
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||||
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||||
body {
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||||
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@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
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<script type="text/x-mathjax-config">
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<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
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<script type="text/x-mathjax-config">
|
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|
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tex2jax: {
|
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inlineMath: [ ['$','$'], ["\\(","\\)"] ],
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|
@ -11766,23 +11762,21 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
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||||
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<body>
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||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
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||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
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||||
<div class="container" id="notebook-container">
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||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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||||
<h1 id="Euler-Problem-38">Euler Problem 38<a class="anchor-link" href="#Euler-Problem-38">¶</a></h1><p>Take the number 192 and multiply it by each of 1, 2, and 3:</p>
|
||||
<h1 id="Euler-Problem-38">Euler Problem 38<a class="anchor-link" href="#Euler-Problem-38">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>Take the number 192 and multiply it by each of 1, 2, and 3:</p>
|
||||
<p>$192 \times 1 = 192$</p>
|
||||
<p>$192 \times 2 = 384$</p>
|
||||
<p>$192 \times 3 = 576$</p>
|
||||
<p>By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3)</p>
|
||||
<p>The same can be achieved by starting with 9 and multiplying by 1, 2, 3, 4, and 5, giving the pandigital, 918273645, which is the concatenated product of 9 and (1,2,3,4,5).</p>
|
||||
<p>What is the largest 1 to 9 pandigital 9-digit number that can be formed as the concatenated product of an integer with (1,2, ... , n) where n > 1?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11791,46 +11785,42 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Let's start by implementing a function which can calculate the concatenated product of a number based on a given $n$.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
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||||
<div class="input">
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||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_concatenated_product_of</span><span class="p">(</span><span class="n">number</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
|
||||
<span class="n">number</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">number</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">number</span> <span class="o">*</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
|
||||
<span class="k">return</span> <span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">number</span> <span class="o">*</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_concatenated_product_of</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"918273645"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_concatenated_product_of</span><span class="p">(</span><span class="mi">192</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"192384576"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_concatenated_product_of</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"918273645"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_concatenated_product_of</span><span class="p">(</span><span class="mi">192</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"192384576"</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
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|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
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|
||||
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|
||||
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|
||||
<p>Next we need a function that can check if a number is pandigital for the numbers from 1 to 9. The idea is that we generate a validation array for the numbers 1 to 9. Each number maps to a field in the array and we initialize the array with zeros. We then iterate over the input number and increment the respective validation field by one. Afterwards we check if each field in the validation array is one in which case the number is n-pandigital. Otherwise, it is not.</p>
|
||||
|
||||
</div>
|
||||
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|
||||
</div>
|
||||
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|
||||
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|
||||
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|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
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|
||||
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|
||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_pandigital</span><span class="p">(</span><span class="n">number</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">9</span><span class="p">):</span>
|
||||
<span class="n">number_str</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">number</span><span class="p">)</span>
|
||||
<span class="n">validation_array</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
|
||||
<span class="k">if</span> <span class="s2">"0"</span> <span class="ow">in</span> <span class="n">number_str</span><span class="p">:</span>
|
||||
<span class="k">if</span> <span class="s2">"0"</span> <span class="ow">in</span> <span class="n">number_str</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">for</span> <span class="n">digit</span> <span class="ow">in</span> <span class="n">number_str</span><span class="p">:</span>
|
||||
<span class="n">validation_array</span><span class="p">[</span><span class="nb">ord</span><span class="p">(</span><span class="n">digit</span><span class="p">)</span> <span class="o">-</span> <span class="mi">49</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
|
||||
|
|
@ -11839,25 +11829,22 @@ div#notebook {
|
|||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">return</span> <span class="kc">True</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"012345678"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"123"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"9"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"123568"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"987"</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"918273645"</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"012345678"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"123"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"9"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"123568"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"987"</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_pandigital</span><span class="p">(</span><span class="s2">"918273645"</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
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|
||||
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|
||||
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|
||||
<p>This example shows how important it is to add little tests even for simple programs like this. The first example is certainly not 1 to 9 pandigital, but because of negative array indexing in Python it qualifies as True. We solved this problem by adding a simple check for a "0" in the number_str.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11866,19 +11853,18 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>The next step is to check whether a number has the potential to become a 1 to 9 pandigital. The algorithm is essentially the same except this time we return False only if a field in the validation array is greater than 1. This means at least one digit occurs multiple times. Otherwise, the array has still potential to become a pandigital number.</p>
|
||||
|
||||
</div>
|
||||
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|
||||
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|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">could_be_pandigital</span><span class="p">(</span><span class="n">number</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">9</span><span class="p">):</span>
|
||||
<span class="n">number_str</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">number</span><span class="p">)</span>
|
||||
<span class="n">validation_array</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
|
||||
<span class="k">if</span> <span class="s2">"0"</span> <span class="ow">in</span> <span class="n">number_str</span><span class="p">:</span>
|
||||
<span class="k">if</span> <span class="s2">"0"</span> <span class="ow">in</span> <span class="n">number_str</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">for</span> <span class="n">digit</span> <span class="ow">in</span> <span class="n">number_str</span><span class="p">:</span>
|
||||
<span class="n">validation_array</span><span class="p">[</span><span class="nb">ord</span><span class="p">(</span><span class="n">digit</span><span class="p">)</span> <span class="o">-</span> <span class="mi">49</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
|
||||
|
|
@ -11887,35 +11873,32 @@ div#notebook {
|
|||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">return</span> <span class="kc">True</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"012345678"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"123"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"9"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"123568"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"987"</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"918273645"</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"98233"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"1223"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"012345678"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"123"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"9"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"123568"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"987"</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span> <span class="o">==</span> <span class="kc">True</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"918273645"</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"98233"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">could_be_pandigital</span><span class="p">(</span><span class="s2">"1223"</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>The next function we want is one which checks if a certain number can actually create a concatenated pandigital product of an integer with (1,2, ... , n) where n > 1. For this function we can use all function which we have implemented to this point. We take a number and do a simple brute force starting with n = 2. We return False once the concatenated product cannot be a pandigital anymore. We use the two given examples as tests.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="prompt input_prompt">In [4]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">can_build_concatenated_pandigital_product</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
|
||||
<span class="n">number_str</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">number</span><span class="p">)</span>
|
||||
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">9</span><span class="p">):</span>
|
||||
|
|
@ -11924,152 +11907,116 @@ div#notebook {
|
|||
<span class="k">return</span> <span class="n">concatenated_product</span>
|
||||
<span class="k">if</span> <span class="n">could_be_pandigital</span><span class="p">(</span><span class="n">concatenated_product</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s2">"If we got here we have a bug."</span><span class="p">)</span>
|
||||
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s2">"If we got here we have a bug."</span><span class="p">)</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">can_build_concatenated_pandigital_product</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"918273645"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">can_build_concatenated_pandigital_product</span><span class="p">(</span><span class="mi">192</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"192384576"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">can_build_concatenated_pandigital_product</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"918273645"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">can_build_concatenated_pandigital_product</span><span class="p">(</span><span class="mi">192</span><span class="p">)</span> <span class="o">==</span> <span class="s2">"192384576"</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>From here on it is just a matter of bruteforcing. We know that the base number of a higher concatenated pandigital product has to start with a 9. Also we know that the base number cannot have more than 5 digits because $10000 \times 1, 10000 \times 2 = 1000020000" has already more digits than a potential solution. This means we create a list of possible solutions first.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [5]:</div>
|
||||
<div class="prompt input_prompt">In [5]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">possible_solutions</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s2">"9"</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="s2">"0"</span> <span class="ow">in</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s2">"9"</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="ow">not</span> <span class="s2">"0"</span> <span class="ow">in</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
|
||||
<span class="k">if</span> <span class="n">could_be_pandigital</span><span class="p">(</span><span class="n">i</span><span class="p">)]</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Number of possible solutions: </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">possible_solutions</span><span class="p">)))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Number of possible solutions: </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">possible_solutions</span><span class="p">)))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>Number of possible solutions: 401
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Since this number is actually small enough we filter all base numbers which can produce a concatenated pandigital product.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [6]:</div>
|
||||
<div class="prompt input_prompt">In [6]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">possible_solutions</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">possible_solutions</span>
|
||||
<span class="k">if</span> <span class="n">can_build_concatenated_pandigital_product</span><span class="p">(</span><span class="n">s</span><span class="p">)]</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Number of possible solutions: </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">possible_solutions</span><span class="p">)))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Number of possible solutions: </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">possible_solutions</span><span class="p">)))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>Number of possible solutions: 4
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Finally, calculate the actual product and print the highest one.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [7]:</div>
|
||||
<div class="prompt input_prompt">In [7]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="nb">map</span><span class="p">(</span><span class="n">can_build_concatenated_pandigital_product</span><span class="p">,</span> <span class="n">possible_solutions</span><span class="p">)))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">932718654</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>932718654
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem039</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,19 +11762,17 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Euler-Problem-39">Euler Problem 39<a class="anchor-link" href="#Euler-Problem-39">¶</a></h1><p>If p is the perimeter of a right angle triangle with integral length sides, {a,b,c}, there are exactly three solutions for p = 120.</p>
|
||||
<h1 id="Euler-Problem-39">Euler Problem 39<a class="anchor-link" href="#Euler-Problem-39">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>If p is the perimeter of a right angle triangle with integral length sides, {a,b,c}, there are exactly three solutions for p = 120.</p>
|
||||
<p>$\{20,48,52\}, \{24,45,51\}, \{30,40,50\}$</p>
|
||||
<p>For which value of p ≤ 1000, is the number of solutions maximised?</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11787,15 +11781,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>We are looking for right angle triangles so Pythagoras' theorem $a^2 + b^2 = c^2$ can be used. Also it must be true that $a <= b <= c$ is true for every solution. Let's start with a function that tests the given examples.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [13]:</div>
|
||||
<div class="prompt input_prompt">In [13]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_right_angle_triangle</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
|
||||
<span class="k">if</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">c</span> <span class="o">!=</span> <span class="n">p</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="kc">False</span>
|
||||
|
|
@ -11808,26 +11801,23 @@ div#notebook {
|
|||
<span class="k">assert</span><span class="p">(</span><span class="n">is_right_angle_triangle</span><span class="p">(</span><span class="o">*</span><span class="n">g</span><span class="p">))</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">is_right_angle_triangle</span><span class="p">(</span><span class="mi">29</span><span class="p">,</span> <span class="mi">41</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">120</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>This seems bruteforceable. Let's try to find the solutions for p = 120.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [16]:</div>
|
||||
<div class="prompt input_prompt">In [16]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">brute_force</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
|
||||
<span class="n">solutions</span> <span class="o">=</span> <span class="p">[(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span> <span class="n">p</span> <span class="o">-</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
|
||||
<span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">p</span> <span class="o">//</span> <span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
|
||||
|
|
@ -11838,99 +11828,73 @@ div#notebook {
|
|||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">brute_force</span><span class="p">(</span><span class="mi">120</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
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||||
|
||||
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|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>[(30, 40, 50), (24, 45, 51), (20, 48, 52)]
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Looks good. Let's try for all $p <= 1000$.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [18]:</div>
|
||||
<div class="prompt input_prompt">In [18]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">solutions</span> <span class="o">=</span> <span class="p">[(</span><span class="nb">len</span><span class="p">(</span><span class="n">brute_force</span><span class="p">(</span><span class="n">p</span><span class="p">)),</span> <span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1001</span><span class="p">)]</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Well, not too fast, but also not too slow. Let's get the max and we have our solution.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [24]:</div>
|
||||
<div class="prompt input_prompt">In [24]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">solutions</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">p</span> <span class="o">==</span> <span class="mi">840</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>840
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem040</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,25 +11762,22 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Champernowne's-constant-(Euler-Problem-40)">Champernowne's constant (Euler Problem 40)<a class="anchor-link" href="#Champernowne's-constant-(Euler-Problem-40)">¶</a></h1><p><a href="https://projecteuler.net/problem=40">https://projecteuler.net/problem=40</a></p>
|
||||
<h1 id="Champernowne's-constant-(Euler-Problem-40)">Champernowne's constant (Euler Problem 40)<a class="anchor-link" href="#Champernowne's-constant-(Euler-Problem-40)">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p><a href="https://projecteuler.net/problem=40">https://projecteuler.net/problem=40</a></p>
|
||||
<p>An irrational decimal fraction is created by concatenating the positive integers:</p>
|
||||
|
||||
<pre><code>0.123456789101112131415161718192021...
|
||||
|
||||
</code></pre>
|
||||
<p>It can be seen that the 12th digit of the fractional part is 1.</p>
|
||||
<p>If dn represents the nth digit of the fractional part, find the value of the following expression.</p>
|
||||
<p>$d_{1} × d_{10} × d_{100} × d_{1000} × d_{10000} × d_{100000} × d_{1000000}$</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11793,15 +11786,14 @@ div#notebook {
|
|||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>I do not see why we cannot hold $8 bytes \times 10^{6} = 8 MB$ in memory.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="prompt input_prompt">In [1]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_irrational_fraction</span><span class="p">(</span><span class="n">digits_max</span><span class="p">):</span>
|
||||
<span class="n">s</span> <span class="o">=</span> <span class="p">[]</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
|
|
@ -11810,81 +11802,63 @@ div#notebook {
|
|||
<span class="n">s</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
|
||||
<span class="n">digits</span> <span class="o">+=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
|
||||
<span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
|
||||
<span class="k">return</span> <span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_irrational_fraction</span><span class="p">(</span><span class="mi">20</span><span class="p">)[</span><span class="mi">12</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"1"</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">get_irrational_fraction</span><span class="p">(</span><span class="mi">20</span><span class="p">)[</span><span class="mi">12</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"1"</span><span class="p">)</span>
|
||||
|
||||
<span class="n">f</span> <span class="o">=</span> <span class="n">get_irrational_fraction</span><span class="p">(</span><span class="mi">1000001</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Okay, we got the fractional. No we get the digits at the different positions and calculate the product. For that purpose we reuse our poduct function from problem 8.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="prompt input_prompt">In [2]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">product</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
|
||||
<span class="kn">from</span> <span class="nn">operator</span> <span class="k">import</span> <span class="n">mul</span>
|
||||
<span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">reduce</span>
|
||||
<span class="k">return</span> <span class="n">reduce</span><span class="p">(</span><span class="n">mul</span><span class="p">,</span> <span class="n">xs</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
<div class="cell border-box-sizing code_cell rendered">
|
||||
<div class="input">
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="prompt input_prompt">In [3]:</div>
|
||||
<div class="inner_cell">
|
||||
<div class="input_area">
|
||||
<div class="input_area">
|
||||
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">s</span> <span class="o">=</span> <span class="n">product</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="p">[</span><span class="n">f</span><span class="p">[</span><span class="mi">10</span><span class="o">**</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">)]))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
|
||||
<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">210</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div class="output_wrapper">
|
||||
<div class="output">
|
||||
|
||||
|
||||
<div class="output_area">
|
||||
|
||||
<div class="prompt"></div>
|
||||
|
||||
|
||||
<div class="output_subarea output_stream output_stdout output_text">
|
||||
<pre>210
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,9 +1,9 @@
|
|||
<!DOCTYPE html>
|
||||
|
||||
<html>
|
||||
<head><meta charset="utf-8" />
|
||||
<head><meta charset="utf-8"/>
|
||||
<title>EulerProblem041</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
|
||||
|
||||
<style type="text/css">
|
||||
/*!
|
||||
*
|
||||
|
|
@ -11714,8 +11714,6 @@ ul.typeahead-list > li > a {
|
|||
.ansi-bold { font-weight: bold; }
|
||||
|
||||
</style>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
/* Overrides of notebook CSS for static HTML export */
|
||||
body {
|
||||
|
|
@ -11741,15 +11739,13 @@ div#notebook {
|
|||
}
|
||||
}
|
||||
</style>
|
||||
|
||||
<!-- Custom stylesheet, it must be in the same directory as the html file -->
|
||||
<link rel="stylesheet" href="custom.css">
|
||||
|
||||
<link href="custom.css" rel="stylesheet"/>
|
||||
<!-- Loading mathjax macro -->
|
||||
<!-- Load mathjax -->
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS_HTML"></script>
|
||||
<!-- MathJax configuration -->
|
||||
<script type="text/x-mathjax-config">
|
||||
MathJax.Hub.Config({
|
||||
tex2jax: {
|
||||
inlineMath: [ ['$','$'], ["\\(","\\)"] ],
|
||||
|
|
@ -11766,16 +11762,15 @@ div#notebook {
|
|||
}
|
||||
});
|
||||
</script>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<!-- End of mathjax configuration --></head>
|
||||
<body>
|
||||
<div tabindex="-1" id="notebook" class="border-box-sizing">
|
||||
<div class="container" id="notebook-container">
|
||||
|
||||
<div class="border-box-sizing" id="notebook" tabindex="-1">
|
||||
<div class="container" id="notebook-container">
|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<h1 id="Pandigital-prime-(Euler-Problem-41)">Pandigital prime (Euler Problem 41)<a class="anchor-link" href="#Pandigital-prime-(Euler-Problem-41)">¶</a></h1>
|
||||
<h1 id="Pandigital-prime-(Euler-Problem-41)">Pandigital prime (Euler Problem 41)<a class="anchor-link" href="#Pandigital-prime-(Euler-Problem-41)">¶</a></h1><p><a href="/euler">Back to overview.</a></p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -11785,28 +11780,21 @@ div#notebook {
|
|||
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<p>We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.</p>
|
||||
<p>What is the largest n-digit pandigital prime that exists?</p>
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||||
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<h1 id="Coded-triangle-numbers-(Euler-Problem-42)">Coded triangle numbers (Euler Problem 42)<a class="anchor-link" href="#Coded-triangle-numbers-(Euler-Problem-42)">¶</a></h1>
|
||||
<h1 id="Coded-triangle-numbers-(Euler-Problem-42)">Coded triangle numbers (Euler Problem 42)<a class="anchor-link" href="#Coded-triangle-numbers-(Euler-Problem-42)">¶</a></h1><p><a href="/euler">Back to overview.</a></p>
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<p>The nth term of the sequence of triangle numbers is given by, $t_n = \frac{1}{2}n(n+1)$; so the first ten triangle numbers are:</p>
|
||||
|
||||
<pre><code>1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
|
||||
|
||||
</code></pre>
|
||||
<p>By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is 19 + 11 + 25 = 55 = $t_{10}$. If the word value is a triangle number then we shall call the word a triangle word.</p>
|
||||
<p>Using words.txt (right click and 'Save Link/Target As...'), a 16K text file containing nearly two-thousand common English words, how many are triangle words? (The file is saved in the same directory as this notebook file as EulerProblem042.txt)</p>
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<h1 id="Sub-string-divisibility-(Euler-Problem-43)">Sub-string divisibility (Euler Problem 43)<a class="anchor-link" href="#Sub-string-divisibility-(Euler-Problem-43)">¶</a></h1>
|
||||
<h1 id="Sub-string-divisibility-(Euler-Problem-43)">Sub-string divisibility (Euler Problem 43)<a class="anchor-link" href="#Sub-string-divisibility-(Euler-Problem-43)">¶</a></h1><p><a href="/euler">Back to overview.</a></p>
|
||||
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@ -11793,28 +11788,21 @@ div#notebook {
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<p>$d_7d_8d_9=728$ is divisible by 13</p>
|
||||
<p>$d_8d_9d_{10} =289$ is divisible by 17</p>
|
||||
<p>Find the sum of all 0 to 9 pandigital numbers with this property.</p>
|
||||
|
||||
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|
||||
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||||
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<title>EulerProblem067</title><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
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||||
<link href="custom.css" rel="stylesheet"/>
|
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<h1 id="Euler-Problem-67">Euler Problem 67<a class="anchor-link" href="#Euler-Problem-67">¶</a></h1><p>By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.</p>
|
||||
|
||||
<h1 id="Euler-Problem-67">Euler Problem 67<a class="anchor-link" href="#Euler-Problem-67">¶</a></h1><p><a href="/euler">Back to overview.</a></p><p>By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.</p>
|
||||
<pre><code> 3
|
||||
7 4
|
||||
2 4 6
|
||||
|
|
@ -11784,7 +11778,6 @@ div#notebook {
|
|||
<p>That is, 3 + 7 + 4 + 9 = 23.</p>
|
||||
<p>Find the maximum total from top to bottom in triangle.txt (right click and 'Save Link/Target As...'), a 15K text file containing a triangle with one-hundred rows.</p>
|
||||
<p>NOTE: This is a much more difficult version of Problem 18. It is not possible to try every route to solve this problem, as there are 299 altogether! If you could check one trillion (1012) routes every second it would take over twenty billion years to check them all. There is an efficient algorithm to solve it. ;o)</p>
|
||||
|
||||
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|
||||
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|
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|
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|
||||
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|
||||
<p>So let's hope our solution is good. I have saved the triangle in a file EulerProblem067.txt in the same directory as this notebook.</p>
|
||||
|
||||
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|
||||
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">find_greatest_path_sum_in_triangle_string</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
|
||||
<span class="kn">from</span> <span class="nn">functools</span> <span class="k">import</span> <span class="n">reduce</span>
|
||||
<span class="n">xss</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">xs</span><span class="o">.</span><span class="n">split</span><span class="p">()))</span> <span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="n">ts</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span> <span class="k">if</span> <span class="n">xs</span><span class="p">]</span>
|
||||
<span class="n">xss</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">xs</span><span class="o">.</span><span class="n">split</span><span class="p">()))</span> <span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="n">ts</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span> <span class="k">if</span> <span class="n">xs</span><span class="p">]</span>
|
||||
<span class="n">xss</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
|
||||
<span class="n">r</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">:</span> <span class="p">[</span><span class="nb">max</span><span class="p">([</span><span class="n">xs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">ys</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">xs</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">ys</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ys</span><span class="p">))]</span>
|
||||
<span class="k">return</span> <span class="n">reduce</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">xss</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">xss</span><span class="p">[</span><span class="mi">0</span><span class="p">])[</span><span class="mi">0</span><span class="p">]</span>
|
||||
|
||||
<span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s1">'EulerProblem067.txt'</span><span class="p">,</span> <span class="s1">'r'</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
|
||||
<span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s1">'EulerProblem067.txt'</span><span class="p">,</span> <span class="s1">'r'</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
|
||||
<span class="n">triangle_string</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">read</span><span class="p">()</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">find_greatest_path_sum_in_triangle_string</span><span class="p">(</span><span class="n">triangle_string</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
|
||||
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|
||||
|
||||
|
||||
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|
||||
<pre>7273
|
||||
</pre>
|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
|
||||
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|
||||
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
|
||||
</div>
|
||||
<div class="inner_cell">
|
||||
<div class="text_cell_render border-box-sizing rendered_html">
|
||||
<p>Well, that was fast.</p>
|
||||
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</body>
|
||||
|
||||
|
||||
|
||||
|
||||
</html>
|
||||
|
|
|
|||
|
|
@ -1,47 +1,48 @@
|
|||
<!doctype html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- Required meta tags -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
|
||||
|
||||
<!-- Bootstrap CSS -->
|
||||
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/css/bootstrap.min.css" integrity="sha384-Gn5384xqQ1aoWXA+058RXPxPg6fy4IWvTNh0E263XmFcJlSAwiGgFAW/dAiS6JXm" crossorigin="anonymous">
|
||||
|
||||
<title>Project Euler Solutions</title>
|
||||
<meta name="description" content="Felix Martin's Homepage">
|
||||
<link rel="shortcut icon" type="image/png" href="https://www.felixm.de/static/mannaz.png"/>
|
||||
<title>Euler - Felix Martin</title>
|
||||
<meta name="author" content="Felix Martin">
|
||||
<link href="https://www.felixm.de/static/template.css" rel="stylesheet">
|
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</head>
|
||||
|
||||
<body>
|
||||
<header>
|
||||
<div class="navbar navbar-dark bg-dark box-shadow">
|
||||
<div class="container d-flex justify-content-between">
|
||||
<a href="#" class="navbar-brand d-flex align-items-center">
|
||||
<strong>My Project Euler Solutions</strong>
|
||||
</a>
|
||||
<div class="container">
|
||||
<div class="row">
|
||||
<div class="col">
|
||||
<h1><a href="https://www.felixm.de">Felix Martin</a></h1>
|
||||
</div>
|
||||
</div>
|
||||
</header>
|
||||
|
||||
<main role="main">
|
||||
<div class="container">
|
||||
|
||||
<div class="row" style="padding-top: 40px;">
|
||||
<div class="row">
|
||||
<div class="col">
|
||||
<table class="table">
|
||||
<h2>Project Euler</h2>
|
||||
</div>
|
||||
</div>
|
||||
<div class="row">
|
||||
<div class="col">
|
||||
<table class="table table-hover table-striped">
|
||||
<thead>
|
||||
<tr>
|
||||
<th scope="col">Name</th>
|
||||
<th scope="col">Completion date</th>
|
||||
<th scope="col">Link</th>
|
||||
<th scope="col">Tags</th>
|
||||
<th style="width: 20%" scope="col">Link</th>
|
||||
<th style="width: 30%" scope="col">Completion date</th>
|
||||
<th style="width: 50%" scope="col">Tags</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 001</td>
|
||||
<td>Tue, 19 Aug 2014, 20:11</td>
|
||||
|
||||
<td><a href="EulerProblem001.html">Problem 001</a></td>
|
||||
<td>Tue, 19 Aug 2014, 20:11</td>
|
||||
<td>
|
||||
|
||||
<kbd>brute force</kbd>
|
||||
|
|
@ -49,10 +50,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 002</td>
|
||||
<td>Tue, 19 Aug 2014, 20:36</td>
|
||||
|
||||
<td><a href="EulerProblem002.html">Problem 002</a></td>
|
||||
<td>Tue, 19 Aug 2014, 20:36</td>
|
||||
<td>
|
||||
|
||||
<kbd>fibonacci</kbd>
|
||||
|
|
@ -60,10 +62,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 003</td>
|
||||
<td>Tue, 19 Aug 2014, 23:35</td>
|
||||
|
||||
<td><a href="EulerProblem003.html">Problem 003</a></td>
|
||||
<td>Tue, 19 Aug 2014, 23:35</td>
|
||||
<td>
|
||||
|
||||
<kbd>prime</kbd>
|
||||
|
|
@ -71,10 +74,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 004</td>
|
||||
<td>Wed, 20 Aug 2014, 13:27</td>
|
||||
|
||||
<td><a href="EulerProblem004.html">Problem 004</a></td>
|
||||
<td>Wed, 20 Aug 2014, 13:27</td>
|
||||
<td>
|
||||
|
||||
<kbd>palindrome</kbd>
|
||||
|
|
@ -88,10 +92,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 005</td>
|
||||
<td>Wed, 20 Aug 2014, 14:32</td>
|
||||
|
||||
<td><a href="EulerProblem005.html">Problem 005</a></td>
|
||||
<td>Wed, 20 Aug 2014, 14:32</td>
|
||||
<td>
|
||||
|
||||
<kbd>reduce</kbd>
|
||||
|
|
@ -101,19 +106,21 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 006</td>
|
||||
<td>Wed, 20 Aug 2014, 15:13</td>
|
||||
|
||||
<td><a href="EulerProblem006.html">Problem 006</a></td>
|
||||
<td>Wed, 20 Aug 2014, 15:13</td>
|
||||
<td>
|
||||
|
||||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 007</td>
|
||||
<td>Wed, 20 Aug 2014, 15:40</td>
|
||||
|
||||
<td><a href="EulerProblem007.html">Problem 007</a></td>
|
||||
<td>Wed, 20 Aug 2014, 15:40</td>
|
||||
<td>
|
||||
|
||||
<kbd>prime</kbd>
|
||||
|
|
@ -127,10 +134,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 008</td>
|
||||
<td>Wed, 20 Aug 2014, 16:03</td>
|
||||
|
||||
<td><a href="EulerProblem008.html">Problem 008</a></td>
|
||||
<td>Wed, 20 Aug 2014, 16:03</td>
|
||||
<td>
|
||||
|
||||
<kbd>product</kbd>
|
||||
|
|
@ -142,10 +150,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 009</td>
|
||||
<td>Wed, 20 Aug 2014, 17:06</td>
|
||||
|
||||
<td><a href="EulerProblem009.html">Problem 009</a></td>
|
||||
<td>Wed, 20 Aug 2014, 17:06</td>
|
||||
<td>
|
||||
|
||||
<kbd>brute force</kbd>
|
||||
|
|
@ -157,10 +166,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 010</td>
|
||||
<td>Wed, 20 Aug 2014, 20:28</td>
|
||||
|
||||
<td><a href="EulerProblem010.html">Problem 010</a></td>
|
||||
<td>Wed, 20 Aug 2014, 20:28</td>
|
||||
<td>
|
||||
|
||||
<kbd>primes</kbd>
|
||||
|
|
@ -172,10 +182,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 011</td>
|
||||
<td>Sun, 31 Aug 2014, 07:39</td>
|
||||
|
||||
<td><a href="EulerProblem011.html">Problem 011</a></td>
|
||||
<td>Sun, 31 Aug 2014, 07:39</td>
|
||||
<td>
|
||||
|
||||
<kbd>brute force</kbd>
|
||||
|
|
@ -191,10 +202,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 012</td>
|
||||
<td>Sun, 31 Aug 2014, 17:07</td>
|
||||
|
||||
<td><a href="EulerProblem012.html">Problem 012</a></td>
|
||||
<td>Sun, 31 Aug 2014, 17:07</td>
|
||||
<td>
|
||||
|
||||
<kbd>triangular</kbd>
|
||||
|
|
@ -212,10 +224,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 013</td>
|
||||
<td>Sun, 31 Aug 2014, 19:03</td>
|
||||
|
||||
<td><a href="EulerProblem013.html">Problem 013</a></td>
|
||||
<td>Sun, 31 Aug 2014, 19:03</td>
|
||||
<td>
|
||||
|
||||
<kbd>brute force</kbd>
|
||||
|
|
@ -225,10 +238,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 014</td>
|
||||
<td>Sun, 31 Aug 2014, 19:45</td>
|
||||
|
||||
<td><a href="EulerProblem014.html">Problem 014</a></td>
|
||||
<td>Sun, 31 Aug 2014, 19:45</td>
|
||||
<td>
|
||||
|
||||
<kbd>collatz</kbd>
|
||||
|
|
@ -240,10 +254,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 015</td>
|
||||
<td>Sun, 31 Aug 2014, 20:05</td>
|
||||
|
||||
<td><a href="EulerProblem015.html">Problem 015</a></td>
|
||||
<td>Sun, 31 Aug 2014, 20:05</td>
|
||||
<td>
|
||||
|
||||
<kbd>factorial</kbd>
|
||||
|
|
@ -253,10 +268,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 016</td>
|
||||
<td>Sun, 31 Aug 2014, 20:49</td>
|
||||
|
||||
<td><a href="EulerProblem016.html">Problem 016</a></td>
|
||||
<td>Sun, 31 Aug 2014, 20:49</td>
|
||||
<td>
|
||||
|
||||
<kbd>c</kbd>
|
||||
|
|
@ -266,10 +282,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 017</td>
|
||||
<td>Sun, 31 Aug 2014, 21:20</td>
|
||||
|
||||
<td><a href="EulerProblem017.html">Problem 017</a></td>
|
||||
<td>Sun, 31 Aug 2014, 21:20</td>
|
||||
<td>
|
||||
|
||||
<kbd>count</kbd>
|
||||
|
|
@ -281,10 +298,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 018</td>
|
||||
<td>Thu, 4 Sep 2014, 20:38</td>
|
||||
|
||||
<td><a href="EulerProblem018.html">Problem 018</a></td>
|
||||
<td>Thu, 4 Sep 2014, 20:38</td>
|
||||
<td>
|
||||
|
||||
<kbd>fold</kbd>
|
||||
|
|
@ -296,10 +314,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 019</td>
|
||||
<td>Fri, 5 Sep 2014, 09:56</td>
|
||||
|
||||
<td><a href="EulerProblem019.html">Problem 019</a></td>
|
||||
<td>Fri, 5 Sep 2014, 09:56</td>
|
||||
<td>
|
||||
|
||||
<kbd>weekdays</kbd>
|
||||
|
|
@ -313,10 +332,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 020</td>
|
||||
<td>Fri, 5 Sep 2014, 10:42</td>
|
||||
|
||||
<td><a href="EulerProblem020.html">Problem 020</a></td>
|
||||
<td>Fri, 5 Sep 2014, 10:42</td>
|
||||
<td>
|
||||
|
||||
<kbd>factorial</kbd>
|
||||
|
|
@ -324,10 +344,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 021</td>
|
||||
<td>Fri, 5 Sep 2014, 14:39</td>
|
||||
|
||||
<td><a href="EulerProblem021.html">Problem 021</a></td>
|
||||
<td>Fri, 5 Sep 2014, 14:39</td>
|
||||
<td>
|
||||
|
||||
<kbd>amicable</kbd>
|
||||
|
|
@ -341,10 +362,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 022</td>
|
||||
<td>Fri, 5 Sep 2014, 15:24</td>
|
||||
|
||||
<td><a href="EulerProblem022.html">Problem 022</a></td>
|
||||
<td>Fri, 5 Sep 2014, 15:24</td>
|
||||
<td>
|
||||
|
||||
<kbd>sorting</kbd>
|
||||
|
|
@ -354,10 +376,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 023</td>
|
||||
<td>Thu, 5 Nov 2015, 14:48</td>
|
||||
|
||||
<td><a href="EulerProblem023.html">Problem 023</a></td>
|
||||
<td>Thu, 5 Nov 2015, 14:48</td>
|
||||
<td>
|
||||
|
||||
<kbd>perfect number</kbd>
|
||||
|
|
@ -369,10 +392,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 024</td>
|
||||
<td>Thu, 5 Nov 2015, 16:04</td>
|
||||
|
||||
<td><a href="EulerProblem024.html">Problem 024</a></td>
|
||||
<td>Thu, 5 Nov 2015, 16:04</td>
|
||||
<td>
|
||||
|
||||
<kbd>permutation</kbd>
|
||||
|
|
@ -380,10 +404,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 025</td>
|
||||
<td>Fri, 5 Sep 2014, 14:57</td>
|
||||
|
||||
<td><a href="EulerProblem025.html">Problem 025</a></td>
|
||||
<td>Fri, 5 Sep 2014, 14:57</td>
|
||||
<td>
|
||||
|
||||
<kbd>fibonacci</kbd>
|
||||
|
|
@ -391,10 +416,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 026</td>
|
||||
<td>Mon, 9 Nov 2015, 22:11</td>
|
||||
|
||||
<td><a href="EulerProblem026.html">Problem 026</a></td>
|
||||
<td>Mon, 9 Nov 2015, 22:11</td>
|
||||
<td>
|
||||
|
||||
<kbd>reciprocal</kbd>
|
||||
|
|
@ -408,10 +434,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 027</td>
|
||||
<td>Mon, 21 Aug 2017, 21:11</td>
|
||||
|
||||
<td><a href="EulerProblem027.html">Problem 027</a></td>
|
||||
<td>Mon, 21 Aug 2017, 21:11</td>
|
||||
<td>
|
||||
|
||||
<kbd>quadratic</kbd>
|
||||
|
|
@ -425,10 +452,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 028</td>
|
||||
<td>Wed, 23 Aug 2017, 15:54</td>
|
||||
|
||||
<td><a href="EulerProblem028.html">Problem 028</a></td>
|
||||
<td>Wed, 23 Aug 2017, 15:54</td>
|
||||
<td>
|
||||
|
||||
<kbd>spiral</kbd>
|
||||
|
|
@ -438,10 +466,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 029</td>
|
||||
<td>Fri, 25 Aug 2017, 10:03</td>
|
||||
|
||||
<td><a href="EulerProblem029.html">Problem 029</a></td>
|
||||
<td>Fri, 25 Aug 2017, 10:03</td>
|
||||
<td>
|
||||
|
||||
<kbd>distinct</kbd>
|
||||
|
|
@ -451,10 +480,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 030</td>
|
||||
<td>Fri, 25 Aug 2017, 12:13</td>
|
||||
|
||||
<td><a href="EulerProblem030.html">Problem 030</a></td>
|
||||
<td>Fri, 25 Aug 2017, 12:13</td>
|
||||
<td>
|
||||
|
||||
<kbd>digit</kbd>
|
||||
|
|
@ -466,10 +496,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 031</td>
|
||||
<td>Fri, 25 Aug 2017, 13:02</td>
|
||||
|
||||
<td><a href="EulerProblem031.html">Problem 031</a></td>
|
||||
<td>Fri, 25 Aug 2017, 13:02</td>
|
||||
<td>
|
||||
|
||||
<kbd>recursion</kbd>
|
||||
|
|
@ -479,10 +510,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 032</td>
|
||||
<td>Wed, 30 Aug 2017, 14:25</td>
|
||||
|
||||
<td><a href="EulerProblem032.html">Problem 032</a></td>
|
||||
<td>Wed, 30 Aug 2017, 14:25</td>
|
||||
<td>
|
||||
|
||||
<kbd>pandigital</kbd>
|
||||
|
|
@ -492,10 +524,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 033</td>
|
||||
<td>Mon, 12 Feb 2018, 17:29</td>
|
||||
|
||||
<td><a href="EulerProblem033.html">Problem 033</a></td>
|
||||
<td>Mon, 12 Feb 2018, 17:29</td>
|
||||
<td>
|
||||
|
||||
<kbd>gcd</kbd>
|
||||
|
|
@ -507,10 +540,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 034</td>
|
||||
<td>Mon, 12 Feb 2018, 17:57</td>
|
||||
|
||||
<td><a href="EulerProblem034.html">Problem 034</a></td>
|
||||
<td>Mon, 12 Feb 2018, 17:57</td>
|
||||
<td>
|
||||
|
||||
<kbd>factorial</kbd>
|
||||
|
|
@ -520,10 +554,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 035</td>
|
||||
<td>Tue, 13 Feb 2018, 09:14</td>
|
||||
|
||||
<td><a href="EulerProblem035.html">Problem 035</a></td>
|
||||
<td>Tue, 13 Feb 2018, 09:14</td>
|
||||
<td>
|
||||
|
||||
<kbd>circular</kbd>
|
||||
|
|
@ -533,10 +568,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 036</td>
|
||||
<td>Fri, 11 May 2018, 02:38</td>
|
||||
|
||||
<td><a href="EulerProblem036.html">Problem 036</a></td>
|
||||
<td>Fri, 11 May 2018, 02:38</td>
|
||||
<td>
|
||||
|
||||
<kbd>palindrome</kbd>
|
||||
|
|
@ -546,10 +582,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 037</td>
|
||||
<td>Sun, 13 May 2018, 16:46</td>
|
||||
|
||||
<td><a href="EulerProblem037.html">Problem 037</a></td>
|
||||
<td>Sun, 13 May 2018, 16:46</td>
|
||||
<td>
|
||||
|
||||
<kbd>primes</kbd>
|
||||
|
|
@ -559,10 +596,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 038</td>
|
||||
<td>Sat, 19 May 2018, 23:45</td>
|
||||
|
||||
<td><a href="EulerProblem038.html">Problem 038</a></td>
|
||||
<td>Sat, 19 May 2018, 23:45</td>
|
||||
<td>
|
||||
|
||||
<kbd>concatenated</kbd>
|
||||
|
|
@ -574,10 +612,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 039</td>
|
||||
<td>Sun, 20 May 2018, 21:16</td>
|
||||
|
||||
<td><a href="EulerProblem039.html">Problem 039</a></td>
|
||||
<td>Sun, 20 May 2018, 21:16</td>
|
||||
<td>
|
||||
|
||||
<kbd>triangle</kbd>
|
||||
|
|
@ -589,10 +628,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 040</td>
|
||||
<td>Sun, 20 May 2018, 00:40</td>
|
||||
|
||||
<td><a href="EulerProblem040.html">Problem 040</a></td>
|
||||
<td>Sun, 20 May 2018, 00:40</td>
|
||||
<td>
|
||||
|
||||
<kbd>champernowne</kbd>
|
||||
|
|
@ -604,10 +644,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td>Problem 041</td>
|
||||
<td></td>
|
||||
|
||||
<tr class="table-warning">
|
||||
|
||||
<td><a href="EulerProblem041.html">Problem 041</a></td>
|
||||
<td></td>
|
||||
<td>
|
||||
|
||||
<kbd>pandigital</kbd>
|
||||
|
|
@ -617,10 +658,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td>Problem 042</td>
|
||||
<td></td>
|
||||
|
||||
<tr class="table-warning">
|
||||
|
||||
<td><a href="EulerProblem042.html">Problem 042</a></td>
|
||||
<td></td>
|
||||
<td>
|
||||
|
||||
<kbd>triangle</kbd>
|
||||
|
|
@ -630,10 +672,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td>Problem 043</td>
|
||||
<td></td>
|
||||
|
||||
<tr class="table-warning">
|
||||
|
||||
<td><a href="EulerProblem043.html">Problem 043</a></td>
|
||||
<td></td>
|
||||
<td>
|
||||
|
||||
<kbd>pandigital</kbd>
|
||||
|
|
@ -643,10 +686,11 @@
|
|||
</td>
|
||||
</tr>
|
||||
|
||||
|
||||
<tr>
|
||||
<td>Problem 067</td>
|
||||
<td>Fri, 5 Sep 2014, 07:36</td>
|
||||
|
||||
<td><a href="EulerProblem067.html">Problem 067</a></td>
|
||||
<td>Fri, 5 Sep 2014, 07:36</td>
|
||||
<td>
|
||||
|
||||
<kbd>reduce</kbd>
|
||||
|
|
@ -660,20 +704,16 @@
|
|||
</table>
|
||||
</div>
|
||||
</div>
|
||||
</div><!-- /.container -->
|
||||
|
||||
|
||||
<div class="row" style="padding-top: 40px;">
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
<!-- FOOTER -->
|
||||
<footer class="container">
|
||||
<p class="float-right"><a href="#">Back to top</a></p>
|
||||
<p>© 2017-2018 Felix Martin ·
|
||||
<a href="#">Impressum</a> ·
|
||||
<a href="https://code.felixm.de">Code</a>
|
||||
</p>
|
||||
</footer>
|
||||
</main>
|
||||
|
||||
|
||||
<!-- Optional JavaScript -->
|
||||
<!-- jQuery first, then Popper.js, then Bootstrap JS -->
|
||||
<script src="https://code.jquery.com/jquery-3.2.1.slim.min.js" integrity="sha384-KJ3o2DKtIkvYIK3UENzmM7KCkRr/rE9/Qpg6aAZGJwFDMVNA/GpGFF93hXpG5KkN" crossorigin="anonymous"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/popper.js/1.12.9/umd/popper.min.js" integrity="sha384-ApNbgh9B+Y1QKtv3Rn7W3mgPxhU9K/ScQsAP7hUibX39j7fakFPskvXusvfa0b4Q" crossorigin="anonymous"></script>
|
||||
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/js/bootstrap.min.js" integrity="sha384-JZR6Spejh4U02d8jOt6vLEHfe/JQGiRRSQQxSfFWpi1MquVdAyjUar5+76PVCmYl" crossorigin="anonymous"></script>
|
||||
</body>
|
||||
</html>
|
||||
|
|
@ -7,6 +7,7 @@ import shutil
|
|||
from operator import itemgetter
|
||||
from collections import namedtuple
|
||||
from os.path import getmtime
|
||||
from bs4 import BeautifulSoup
|
||||
import json
|
||||
|
||||
|
||||
|
|
@ -56,6 +57,28 @@ def convert_solutions_to_html(solutions):
|
|||
if not os.path.isfile(html) or getmtime(html) < getmtime(s.ipynb):
|
||||
args = ["jupyter-nbconvert", s.ipynb, "--output-dir=html"]
|
||||
subprocess.call(args)
|
||||
post_process(html)
|
||||
|
||||
|
||||
def post_process(html):
|
||||
|
||||
def tag(t, string="", children=[], **attrs):
|
||||
t = BeautifulSoup("", 'html.parser').new_tag(t, **attrs)
|
||||
if string:
|
||||
t.string = string
|
||||
for c in children:
|
||||
t.append(c)
|
||||
return t
|
||||
|
||||
with open(html, 'r') as f:
|
||||
soup = BeautifulSoup(f.read(), 'html.parser')
|
||||
|
||||
soup_h1 = soup.find("h1")
|
||||
soup_p = tag("p", children=[tag("a", href="/euler", string="Back to overview.")])
|
||||
soup_h1.insert_after(soup_p)
|
||||
|
||||
with open(html, 'w') as f:
|
||||
f.write(str(soup))
|
||||
|
||||
|
||||
def ship_to_failx():
|
||||
|
|
|
|||
|
|
@ -1,47 +1,50 @@
|
|||
<!doctype html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- Required meta tags -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
|
||||
|
||||
<!-- Bootstrap CSS -->
|
||||
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/css/bootstrap.min.css" integrity="sha384-Gn5384xqQ1aoWXA+058RXPxPg6fy4IWvTNh0E263XmFcJlSAwiGgFAW/dAiS6JXm" crossorigin="anonymous">
|
||||
|
||||
<title>Project Euler Solutions</title>
|
||||
<meta name="description" content="Felix Martin's Homepage">
|
||||
<link rel="shortcut icon" type="image/png" href="https://www.felixm.de/static/mannaz.png"/>
|
||||
<title>Euler - Felix Martin</title>
|
||||
<meta name="author" content="Felix Martin">
|
||||
<link href="https://www.felixm.de/static/template.css" rel="stylesheet">
|
||||
</head>
|
||||
|
||||
<body>
|
||||
<header>
|
||||
<div class="navbar navbar-dark bg-dark box-shadow">
|
||||
<div class="container d-flex justify-content-between">
|
||||
<a href="#" class="navbar-brand d-flex align-items-center">
|
||||
<strong>My Project Euler Solutions</strong>
|
||||
</a>
|
||||
<div class="container">
|
||||
<div class="row">
|
||||
<div class="col">
|
||||
<h1><a href="https://www.felixm.de">Felix Martin</a></h1>
|
||||
</div>
|
||||
</div>
|
||||
</header>
|
||||
|
||||
<main role="main">
|
||||
<div class="container">
|
||||
|
||||
<div class="row" style="padding-top: 40px;">
|
||||
<div class="row">
|
||||
<div class="col">
|
||||
<table class="table">
|
||||
<h2>Project Euler</h2>
|
||||
</div>
|
||||
</div>
|
||||
<div class="row">
|
||||
<div class="col">
|
||||
<table class="table table-hover table-striped">
|
||||
<thead>
|
||||
<tr>
|
||||
<th scope="col">Name</th>
|
||||
<th scope="col">Completion date</th>
|
||||
<th scope="col">Link</th>
|
||||
<th scope="col">Tags</th>
|
||||
<th style="width: 20%" scope="col">Link</th>
|
||||
<th style="width: 30%" scope="col">Completion date</th>
|
||||
<th style="width: 50%" scope="col">Tags</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
{% for s in solutions %}
|
||||
{% if not s.metadata.completion_date %}
|
||||
<tr class="table-warning">
|
||||
{% else %}
|
||||
<tr>
|
||||
<td>{{s.name}}</td>
|
||||
<td>{{s.metadata.completion_date}}</td>
|
||||
{% endif %}
|
||||
<td><a href="{{s.html}}">{{s.name}}</a></td>
|
||||
<td>{{s.metadata.completion_date}}</td>
|
||||
<td>
|
||||
{% for t in s.metadata.tags %}
|
||||
<kbd>{{t}}</kbd>
|
||||
|
|
@ -53,20 +56,16 @@
|
|||
</table>
|
||||
</div>
|
||||
</div>
|
||||
</div><!-- /.container -->
|
||||
|
||||
|
||||
<div class="row" style="padding-top: 40px;">
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
<!-- FOOTER -->
|
||||
<footer class="container">
|
||||
<p class="float-right"><a href="#">Back to top</a></p>
|
||||
<p>© 2017-2018 Felix Martin ·
|
||||
<a href="#">Impressum</a> ·
|
||||
<a href="https://code.felixm.de">Code</a>
|
||||
</p>
|
||||
</footer>
|
||||
</main>
|
||||
|
||||
|
||||
<!-- Optional JavaScript -->
|
||||
<!-- jQuery first, then Popper.js, then Bootstrap JS -->
|
||||
<script src="https://code.jquery.com/jquery-3.2.1.slim.min.js" integrity="sha384-KJ3o2DKtIkvYIK3UENzmM7KCkRr/rE9/Qpg6aAZGJwFDMVNA/GpGFF93hXpG5KkN" crossorigin="anonymous"></script>
|
||||
<script src="https://cdnjs.cloudflare.com/ajax/libs/popper.js/1.12.9/umd/popper.min.js" integrity="sha384-ApNbgh9B+Y1QKtv3Rn7W3mgPxhU9K/ScQsAP7hUibX39j7fakFPskvXusvfa0b4Q" crossorigin="anonymous"></script>
|
||||
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/js/bootstrap.min.js" integrity="sha384-JZR6Spejh4U02d8jOt6vLEHfe/JQGiRRSQQxSfFWpi1MquVdAyjUar5+76PVCmYl" crossorigin="anonymous"></script>
|
||||
</body>
|
||||
</html>
|
||||
|
|
|
|||
Loading…
Reference in New Issue