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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>
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<pre><code> 3
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7 4
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2 4 6
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8 5 9 3</code></pre>
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<p>That is, 3 + 7 + 4 + 9 = 23.</p>
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<p>Find the maximum total from top to bottom of the triangle below:</p>
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<pre><code>75
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95 64
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17 47 82
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63 66 04 68 89 53 67 30 73 16 69 87 40 31
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04 62 98 27 23 09 70 98 73 93 38 53 60 04 23</code></pre>
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<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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<p>This is incredibly simple we simply from bottom to top choosing the higher value for each mini-tree.</p>
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<p>For example,</p>
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<pre><code> 95 64
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17 47 82</code></pre>
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<p>will become:</p>
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<pre><code> 142 146</code></pre>
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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=" 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>
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<span class="s2">95 64</span>
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<span class="s2">91 71 52 38 17 14 91 43 58 50 27 29 48</span>
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<span class="s2">63 66 04 68 89 53 67 30 73 16 69 87 40 31</span>
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<span class="s2">04 62 98 27 23 09 70 98 73 93 38 53 60 04 23</span>
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<span class="s2">"""</span>
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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>
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<span class="sd">""" xs is lower row and ys is upper row """</span>
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<span class="sd">""" xs is lower row and ys is upper row """</span>
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<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>
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<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>
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<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>
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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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<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>
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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>
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<span class="n">xss</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
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<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="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>
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<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>
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<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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<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>
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<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>
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<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>
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<span class="n">xss</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
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<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>
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<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>
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<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>
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<pre>1074
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