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Felix Martin
2018-12-22 18:22:38 -05:00
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<p><a href="https://projecteuler.net/problem=43">https://projecteuler.net/problem=43</a></p>
<p>The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property.</p>
<p>Let $d_1$ be the 1st digit, $d_2$ be the 2nd digit, and so on. In this way, we note the following:</p>
<p>$d_2d_3d_4$=406 is divisible by 2</p>
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<p>We write a function that tests for the sub-string divisibility property. Then we simply brute force.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">is_sub_string_divisible</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="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">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">4</span><span class="p">])</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">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">2</span><span class="p">:</span><span class="mi">5</span><span class="p">])</span> <span class="o">%</span> <span class="mi">3</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">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">s</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="o">%</span> <span class="mi">5</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">if</span> <span class="nb">int</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="mi">7</span><span class="p">])</span> <span class="o">%</span> <span class="mi">7</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">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">s</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="o">%</span> <span class="mi">11</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">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">6</span><span class="p">:</span><span class="mi">9</span><span class="p">])</span> <span class="o">%</span> <span class="mi">13</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">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">7</span><span class="p">:</span><span class="mi">10</span><span class="p">])</span> <span class="o">%</span> <span class="mi">17</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">return</span> <span class="kc">True</span>
<span class="k">assert</span><span class="p">(</span><span class="n">is_sub_string_divisible</span><span class="p">(</span><span class="s2">"1406357289"</span><span class="p">))</span>
<span class="k">assert</span><span class="p">(</span><span class="n">is_sub_string_divisible</span><span class="p">(</span><span class="s2">"1406357298"</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="kn">import</span> <span class="nn">itertools</span>
<span class="n">s</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</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">p</span><span class="p">))</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">itertools</span><span class="o">.</span><span class="n">permutations</span><span class="p">(</span><span class="s2">"0123456789"</span><span class="p">)</span> <span class="k">if</span> <span class="n">is_sub_string_divisible</span><span class="p">(</span><span class="n">p</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">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">16695334890</span><span class="p">)</span>
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<pre>16695334890
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<p>I just want to write my own permutations functions to see that I am able to do it quickly. We pick each element once and combine it with the permutations for the remaining items:</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">permutations</span><span class="p">(</span><span class="n">items</span><span class="p">):</span>
<span class="n">items</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">items</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">items</span><span class="p">:</span>
<span class="k">return</span> <span class="p">[[]]</span>
<span class="n">perms</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">items</span><span class="p">:</span>
<span class="n">items_new</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">items</span><span class="p">)</span>
<span class="n">items_new</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
<span class="n">permutations_new</span> <span class="o">=</span> <span class="n">permutations</span><span class="p">(</span><span class="n">items_new</span><span class="p">)</span>
<span class="k">for</span> <span class="n">perm</span> <span class="ow">in</span> <span class="n">permutations_new</span><span class="p">:</span>
<span class="n">perms</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">item</span><span class="p">]</span> <span class="o">+</span> <span class="n">perm</span><span class="p">)</span>
<span class="k">return</span> <span class="n">perms</span>
<span class="k">assert</span><span class="p">(</span><span class="n">permutations</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">permutations</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">1</span><span class="p">]])</span>
<span class="k">assert</span><span class="p">(</span><span class="n">permutations</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="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="p">[</span><span class="mi">2</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">permutations</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="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="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</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="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</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="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
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