Finished problem 39 and added some new ones.
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<p>$\{20,48,52\}, \{24,45,51\}, \{30,40,50\}$</p>
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<p>For which value of p ≤ 1000, is the number of solutions maximised?</p>
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<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>
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<div class=" highlight hl-ipython3"><pre><span></span>
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<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>
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<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>
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<span class="k">return</span> <span class="kc">False</span>
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<span class="k">if</span> <span class="n">a</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">b</span><span class="o">**</span><span class="mi">2</span> <span class="o">!=</span> <span class="n">c</span><span class="o">**</span><span class="mi">2</span><span class="p">:</span>
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<span class="k">return</span> <span class="kc">False</span>
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<span class="k">return</span> <span class="kc">True</span>
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<span class="n">given</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">48</span><span class="p">,</span> <span class="mi">52</span><span class="p">,</span> <span class="mi">120</span><span class="p">),</span> <span class="p">(</span><span class="mi">24</span><span class="p">,</span> <span class="mi">45</span><span class="p">,</span> <span class="mi">51</span><span class="p">,</span> <span class="mi">120</span><span class="p">),</span> <span class="p">(</span><span class="mi">30</span><span class="p">,</span> <span class="mi">40</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">120</span><span class="p">)]</span>
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<span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">given</span><span class="p">:</span>
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<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>
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<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>
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<p>This seems bruteforceable. Let's try to find the solutions for p = 120.</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">p</span><span class="p">):</span>
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<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>
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<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>
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<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="n">b</span><span class="p">)</span>
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<span class="k">if</span> <span class="n">a</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">b</span> <span class="o">==</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="o">*</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>
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<span class="p">]</span>
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<span class="k">return</span> <span class="n">solutions</span>
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<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>
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<pre>[(30, 40, 50), (24, 45, 51), (20, 48, 52)]
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<p>Looks good. Let's try for all $p <= 1000$.</p>
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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="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>
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<p>Well, not too fast, but also not too slow. Let's get the max and we have our solution.</p>
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<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>
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<span class="nb">print</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
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<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>
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<pre>840
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@@ -576,7 +576,7 @@
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<tr>
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<td>Problem 039</td>
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<td></td>
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<td>Sun, 20 May 2018, 21:16</td>
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<td><a href="EulerProblem039.html">Problem 039</a></td>
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<td>
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@@ -604,6 +604,45 @@
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</td>
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</tr>
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<tr>
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<td>Problem 041</td>
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<td></td>
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<td><a href="EulerProblem041.html">Problem 041</a></td>
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<td>
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<kbd>pandigital</kbd>
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<kbd>prime</kbd>
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</tr>
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<tr>
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<td>Problem 042</td>
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<td></td>
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<td><a href="EulerProblem042.html">Problem 042</a></td>
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<td>
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<kbd>triangle</kbd>
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<kbd>words</kbd>
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<tr>
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<td>Problem 043</td>
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<td></td>
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<td><a href="EulerProblem043.html">Problem 043</a></td>
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<td>
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<kbd>pandigital</kbd>
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<kbd>divisibility</kbd>
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<tr>
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<td>Problem 067</td>
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<td>Fri, 5 Sep 2014, 07:36</td>
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