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Felix Martin
2019-01-06 00:50:37 -05:00
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commit 550a1479f0
7 changed files with 570 additions and 12 deletions
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ipython/html/EulerProblem060.html
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@@ -11786,6 +11786,144 @@ div#notebook {
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<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">combinations</span>
<span class="k">def</span> <span class="nf">concatenate_all</span><span class="p">(</span><span class="n">numbers</span><span class="p">):</span>
<span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">combinations</span><span class="p">(</span><span class="n">numbers</span><span class="p">,</span> <span class="mi">2</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="n">a</span><span class="p">,</span> <span class="n">b</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="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">])</span>
<span class="k">return</span> <span class="n">result</span>
<span class="nb">print</span><span class="p">(</span><span class="n">concatenate_all</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">109</span><span class="p">,</span> <span class="mi">673</span><span class="p">]))</span>
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<pre>[[3, 7], [7, 3], [3, 109], [109, 3], [3, 673], [673, 3], [7, 109], [109, 7], [7, 673], [673, 7], [109, 673], [673, 109]]
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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>
<span class="k">while</span> <span class="n">prospects</span><span class="p">:</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">prospects</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">prospects</span> <span class="o">=</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="n">prospects</span> <span class="k">if</span> <span class="n">x</span> <span class="o">%</span> <span class="n">p</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">]</span>
<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">if</span> <span class="n">p</span> <span class="o">*</span> <span class="n">p</span> <span class="o">&gt;</span> <span class="n">limit</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">primes</span> <span class="o">+=</span> <span class="n">prospects</span>
<span class="k">return</span> <span class="n">primes</span>
<span class="n">primes</span> <span class="o">=</span> <span class="n">sieve_of_eratosthenes</span><span class="p">(</span><span class="mi">100000</span><span class="p">)</span>
<span class="n">primes_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">primes</span><span class="p">)</span>
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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="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="kc">False</span>
<span class="k">while</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">n</span> <span class="o">//=</span> <span class="mi">2</span>
<span class="n">f</span> <span class="o">=</span> <span class="mi">3</span>
<span class="k">while</span> <span class="n">f</span> <span class="o">*</span> <span class="n">f</span> <span class="o">&lt;=</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="n">f</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="kc">False</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">f</span> <span class="o">+=</span> <span class="mi">2</span>
<span class="k">return</span> <span class="kc">True</span>
<span class="k">def</span> <span class="nf">concatentation_is_prime</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">ab</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">a</span><span class="p">)</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
<span class="n">ba</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">b</span><span class="p">)</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
<span class="k">if</span> <span class="n">is_prime</span><span class="p">(</span><span class="n">ab</span><span class="p">)</span> <span class="ow">and</span> <span class="n">is_prime</span><span class="p">(</span><span class="n">ba</span><span class="p">):</span>
<span class="k">return</span> <span class="kc">True</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="kc">False</span>
<span class="k">assert</span><span class="p">(</span><span class="n">concatentation_is_prime</span><span class="p">(</span><span class="mi">673</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="k">assert</span><span class="p">(</span><span class="n">concatentation_is_prime</span><span class="p">(</span><span class="mi">673</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">potentials</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">new_potentials</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">done</span> <span class="o">=</span> <span class="kc">False</span>
<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">primes</span><span class="p">:</span>
<span class="k">if</span> <span class="n">done</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">potentials</span> <span class="o">+=</span> <span class="n">new_potentials</span>
<span class="n">new_potentials</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">potential</span> <span class="ow">in</span> <span class="n">potentials</span><span class="p">:</span>
<span class="n">all_concatenations_prime</span> <span class="o">=</span> <span class="kc">True</span>
<span class="k">for</span> <span class="n">prime</span> <span class="ow">in</span> <span class="n">potential</span><span class="p">:</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">concatentation_is_prime</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">prime</span><span class="p">):</span>
<span class="n">all_concatenations_prime</span> <span class="o">=</span> <span class="kc">False</span>
<span class="k">break</span>
<span class="k">if</span> <span class="n">all_concatenations_prime</span><span class="p">:</span>
<span class="n">new_potential</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">potential</span><span class="p">)</span>
<span class="n">new_potential</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">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">new_potential</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">4</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="n">new_potential</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">new_potential</span><span class="p">)</span>
<span class="n">done</span> <span class="o">=</span> <span class="kc">True</span>
<span class="k">break</span>
<span class="n">new_potentials</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">new_potential</span><span class="p">)</span>
<span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">15</span><span class="p">:</span>
<span class="n">potentials</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="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">26033</span><span class="p">)</span>
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<pre>[13, 5197, 5701, 6733, 8389]
26033
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@@ -11796,6 +11796,100 @@ Find the sum of the only ordered set of six cyclic 4-digit numbers for which eac
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_four_digit_numbers</span><span class="p">(</span><span class="n">function</span><span class="p">):</span>
<span class="n">r</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">function</span>
<span class="k">while</span> <span class="n">f</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">10000</span><span class="p">:</span>
<span class="k">if</span> <span class="n">f</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">999</span><span class="p">:</span>
<span class="n">r</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</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="n">r</span>
<span class="n">triangles</span> <span class="o">=</span> <span class="n">get_four_digit_numbers</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">n</span> <span class="o">*</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="p">)</span>
<span class="n">squares</span> <span class="o">=</span> <span class="n">get_four_digit_numbers</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</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="n">pentas</span> <span class="o">=</span> <span class="n">get_four_digit_numbers</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="mi">3</span> <span class="o">*</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="p">)</span>
<span class="n">hexas</span> <span class="o">=</span> <span class="n">get_four_digit_numbers</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">heptas</span> <span class="o">=</span> <span class="n">get_four_digit_numbers</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">3</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">octas</span> <span class="o">=</span> <span class="n">get_four_digit_numbers</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">))</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_cyclic</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="nb">str</span><span class="p">(</span><span class="n">a</span><span class="p">)[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">==</span> <span class="nb">str</span><span class="p">(</span><span class="n">b</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">is_cyclic</span><span class="p">(</span><span class="mi">3328</span><span class="p">,</span> <span class="mi">2877</span><span class="p">))</span>
<span class="k">assert</span><span class="p">(</span><span class="n">is_cyclic</span><span class="p">(</span><span class="mi">3329</span><span class="p">,</span> <span class="mi">2877</span><span class="p">)</span> <span class="o">==</span> <span class="kc">False</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">search_solution</span><span class="p">(</span><span class="n">aggregator</span><span class="p">,</span> <span class="n">polygonals</span><span class="p">):</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">polygonals</span><span class="p">:</span>
<span class="k">if</span> <span class="n">is_cyclic</span><span class="p">(</span><span class="n">aggregator</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">aggregator</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
<span class="k">return</span> <span class="n">aggregator</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="p">[]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">aggregator</span><span class="p">:</span>
<span class="k">for</span> <span class="n">polygonal</span> <span class="ow">in</span> <span class="n">polygonals</span><span class="p">:</span>
<span class="k">for</span> <span class="n">number</span> <span class="ow">in</span> <span class="n">polygonal</span><span class="p">:</span>
<span class="n">aggregator</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">number</span><span class="p">)</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">search_solution</span><span class="p">(</span><span class="n">aggregator</span><span class="p">,</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">polygonals</span> <span class="k">if</span> <span class="n">p</span> <span class="o">!=</span> <span class="n">polygonal</span><span class="p">])</span>
<span class="k">if</span> <span class="n">s</span><span class="p">:</span>
<span class="k">return</span> <span class="n">s</span>
<span class="n">aggregator</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">for</span> <span class="n">polygonal</span> <span class="ow">in</span> <span class="n">polygonals</span><span class="p">:</span>
<span class="k">for</span> <span class="n">number</span> <span class="ow">in</span> <span class="n">polygonal</span><span class="p">:</span>
<span class="k">if</span> <span class="n">is_cyclic</span><span class="p">(</span><span class="n">aggregator</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">number</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">number</span> <span class="ow">in</span> <span class="n">aggregator</span><span class="p">:</span>
<span class="n">aggregator</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">number</span><span class="p">)</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">search_solution</span><span class="p">(</span><span class="n">aggregator</span><span class="p">,</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">polygonals</span> <span class="k">if</span> <span class="n">p</span> <span class="o">!=</span> <span class="n">polygonal</span><span class="p">])</span>
<span class="k">if</span> <span class="n">s</span><span class="p">:</span>
<span class="k">return</span> <span class="n">s</span>
<span class="n">aggregator</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">return</span> <span class="p">[]</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">search_solution</span><span class="p">([],</span> <span class="p">[</span><span class="n">triangles</span><span class="p">,</span> <span class="n">squares</span><span class="p">,</span> <span class="n">pentas</span><span class="p">,</span> <span class="n">hexas</span><span class="p">,</span> <span class="n">heptas</span><span class="p">,</span> <span class="n">octas</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="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="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">28684</span><span class="p">)</span>
</pre></div>
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<pre>[8256, 5625, 2512, 1281, 8128, 2882]
28684
</pre>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">cubes</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="o">*</span> <span class="n">n</span> <span class="o">*</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">5000</span><span class="p">)]</span>
<span class="n">cubes_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">cubes</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">solutions</span> <span class="o">=</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">1</span><span class="p">,</span> <span class="mi">10000</span><span class="p">):</span>
<span class="n">cube</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="n">n</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">key</span> <span class="o">=</span> <span class="s2">""</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">cube</span><span class="p">)))</span>
<span class="n">solutions</span><span class="p">[</span><span class="n">key</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cube</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">solutions</span><span class="p">[</span><span class="n">key</span><span class="p">])</span> <span class="o">&gt;</span> <span class="mi">4</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="n">solutions</span><span class="p">[</span><span class="n">key</span><span class="p">])</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">solutions</span><span class="p">[</span><span class="n">key</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
<span class="k">break</span>
<span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
<span class="n">solutions</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">cube</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">127035954683</span><span class="p">)</span>
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<pre>[127035954683, 352045367981, 373559126408, 569310543872, 589323567104]
127035954683
</pre>
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@@ -953,32 +953,42 @@
</tr>
<tr class="table-warning">
<tr>
<td><a href="EulerProblem060.html">Problem 060</a></td>
<td></td>
<td>Sun, 6 Jan 2019, 05:00</td>
<td>
<kbd>prime</kbd>
<kbd>pairs</kbd>
</td>
</tr>
<tr class="table-warning">
<tr>
<td><a href="EulerProblem061.html">Problem 061</a></td>
<td></td>
<td>Sun, 6 Jan 2019, 05:32</td>
<td>
<kbd>search</kbd>
<kbd>cyclic</kbd>
</td>
</tr>
<tr class="table-warning">
<tr>
<td><a href="EulerProblem062.html">Problem 062</a></td>
<td></td>
<td>Sun, 6 Jan 2019, 05:47</td>
<td>
<kbd>cubes</kbd>
</td>
</tr>