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<div class="section" id="advent-of-code-2017">
<h1>Advent of Code 2017<a class="headerlink" href="#advent-of-code-2017" title="Permalink to this headline"></a></h1>
<div class="section" id="december-2nd">
<h2>December 2nd<a class="headerlink" href="#december-2nd" title="Permalink to this headline"></a></h2>
<p>For each row, determine the difference between the largest value and the
smallest value; the checksum is the sum of all of these differences.</p>
<p>For example, given the following spreadsheet:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">5</span> <span class="mi">1</span> <span class="mi">9</span> <span class="mi">5</span>
<span class="mi">7</span> <span class="mi">5</span> <span class="mi">3</span>
<span class="mi">2</span> <span class="mi">4</span> <span class="mi">6</span> <span class="mi">8</span>
</pre></div>
</div>
<ul class="simple">
<li>The first row’s largest and smallest values are 9 and 1, and their
difference is 8.</li>
<li>The second row’s largest and smallest values are 7 and 3, and their
difference is 4.</li>
<li>The third row’s difference is 6.</li>
</ul>
<p>In this example, the spreadsheet’s checksum would be 8 + 4 + 6 = 18.</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">notebook_preamble</span> <span class="k">import</span> <span class="n">J</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">define</span>
</pre></div>
</div>
<p>I’ll assume the input is a Joy sequence of sequences of integers.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[[</span><span class="mi">5</span> <span class="mi">1</span> <span class="mi">9</span> <span class="mi">5</span><span class="p">]</span>
<span class="p">[</span><span class="mi">7</span> <span class="mi">5</span> <span class="mi">3</span><span class="p">]</span>
<span class="p">[</span><span class="mi">2</span> <span class="mi">4</span> <span class="mi">6</span> <span class="mi">8</span><span class="p">]]</span>
</pre></div>
</div>
<p>So, obviously, the initial form will be a <code class="docutils literal notranslate"><span class="pre">step</span></code> function:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">AoC2017</span><span class="o">.</span><span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span> <span class="n">swap</span> <span class="p">[</span><span class="n">F</span> <span class="o">+</span><span class="p">]</span> <span class="n">step</span>
</pre></div>
</div>
<p>This function <code class="docutils literal notranslate"><span class="pre">F</span></code> must get the <code class="docutils literal notranslate"><span class="pre">max</span></code> and <code class="docutils literal notranslate"><span class="pre">min</span></code> of a row of numbers
and subtract. We can define a helper function <code class="docutils literal notranslate"><span class="pre">maxmin</span></code> which does
this:</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;maxmin == [max] [min] cleave&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;[1 2 3] maxmin&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">3</span> <span class="mi">1</span>
</pre></div>
</div>
<p>Then <code class="docutils literal notranslate"><span class="pre">F</span></code> just does that then subtracts the min from the max:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">F</span> <span class="o">==</span> <span class="n">maxmin</span> <span class="o">-</span>
</pre></div>
</div>
<p>So:</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;AoC2017.2 == [maxmin - +] step_zero&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;&#39;&#39;</span>
<span class="s1">[[5 1 9 5]</span>
<span class="s1"> [7 5 3]</span>
<span class="s1"> [2 4 6 8]] AoC2017.2</span>
<span class="s1">&#39;&#39;&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">18</span>
</pre></div>
</div>
<p>…find the only two numbers in each row where one evenly divides the
other - that is, where the result of the division operation is a whole
number. They would like you to find those numbers on each line, divide
them, and add up each line’s result.</p>
<p>For example, given the following spreadsheet:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mi">5</span> <span class="mi">9</span> <span class="mi">2</span> <span class="mi">8</span>
<span class="mi">9</span> <span class="mi">4</span> <span class="mi">7</span> <span class="mi">3</span>
<span class="mi">3</span> <span class="mi">8</span> <span class="mi">6</span> <span class="mi">5</span>
</pre></div>
</div>
<ul class="simple">
<li>In the first row, the only two numbers that evenly divide are 8 and
2; the result of this division is 4.</li>
<li>In the second row, the two numbers are 9 and 3; the result is 3.</li>
<li>In the third row, the result is 2.</li>
</ul>
<p>In this example, the sum of the results would be 4 + 3 + 2 = 9.</p>
<p>What is the sum of each row’s result in your puzzle input?</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;[5 9 2 8] sort reverse&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">9</span> <span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span>
</pre></div>
</div>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;[9 8 5 2] uncons [swap [divmod] cons] dupdip&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="p">[</span><span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">9</span> <span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="n">uncons</span> <span class="p">[</span><span class="n">swap</span> <span class="p">[</span><span class="nb">divmod</span><span class="p">]</span> <span class="n">cons</span> <span class="n">F</span><span class="p">]</span> <span class="n">dupdip</span> <span class="n">G</span>
<span class="p">[</span><span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="n">F</span> <span class="p">[</span><span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="n">G</span>
</pre></div>
</div>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">&#39;[8 5 2] [9 divmod] [uncons swap] dip dup [i not] dip&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="o">.</span> <span class="p">[</span><span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="p">[</span><span class="n">uncons</span> <span class="n">swap</span><span class="p">]</span> <span class="n">dip</span> <span class="n">dup</span> <span class="p">[</span><span class="n">i</span> <span class="ow">not</span><span class="p">]</span> <span class="n">dip</span>
<span class="p">[</span><span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="p">[</span><span class="n">uncons</span> <span class="n">swap</span><span class="p">]</span> <span class="n">dip</span> <span class="n">dup</span> <span class="p">[</span><span class="n">i</span> <span class="ow">not</span><span class="p">]</span> <span class="n">dip</span>
<span class="p">[</span><span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">uncons</span> <span class="n">swap</span><span class="p">]</span> <span class="n">dip</span> <span class="n">dup</span> <span class="p">[</span><span class="n">i</span> <span class="ow">not</span><span class="p">]</span> <span class="n">dip</span>
<span class="p">[</span><span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="p">[</span><span class="n">uncons</span> <span class="n">swap</span><span class="p">]</span> <span class="o">.</span> <span class="n">dip</span> <span class="n">dup</span> <span class="p">[</span><span class="n">i</span> <span class="ow">not</span><span class="p">]</span> <span class="n">dip</span>
<span class="p">[</span><span class="mi">8</span> <span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">uncons</span> <span class="n">swap</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="n">dup</span> <span class="p">[</span><span class="n">i</span> <span class="ow">not</span><span class="p">]</span> <span class="n">dip</span>
<span class="mi">8</span> <span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="o">.</span> <span class="n">swap</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="n">dup</span> <span class="p">[</span><span class="n">i</span> <span class="ow">not</span><span class="p">]</span> <span class="n">dip</span>
<span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="mi">8</span> <span class="o">.</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="n">dup</span> <span class="p">[</span><span class="n">i</span> <span class="ow">not</span><span class="p">]</span> <span class="n">dip</span>
<span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="mi">8</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="o">.</span> <span class="n">dup</span> <span class="p">[</span><span class="n">i</span> <span class="ow">not</span><span class="p">]</span> <span class="n">dip</span>
<span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="mi">8</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="o">.</span> <span class="p">[</span><span class="n">i</span> <span class="ow">not</span><span class="p">]</span> <span class="n">dip</span>
<span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="mi">8</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="p">[</span><span class="n">i</span> <span class="ow">not</span><span class="p">]</span> <span class="o">.</span> <span class="n">dip</span>
<span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="mi">8</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="o">.</span> <span class="n">i</span> <span class="ow">not</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span>
<span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="mi">8</span> <span class="o">.</span> <span class="mi">9</span> <span class="nb">divmod</span> <span class="ow">not</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span>
<span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="mi">8</span> <span class="mi">9</span> <span class="o">.</span> <span class="nb">divmod</span> <span class="ow">not</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span>
<span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="mi">1</span> <span class="mi">1</span> <span class="o">.</span> <span class="ow">not</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span>
<span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="mi">1</span> <span class="kc">False</span> <span class="o">.</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span>
<span class="p">[</span><span class="mi">5</span> <span class="mi">2</span><span class="p">]</span> <span class="mi">1</span> <span class="kc">False</span> <span class="p">[</span><span class="mi">9</span> <span class="nb">divmod</span><span class="p">]</span> <span class="o">.</span>
</pre></div>
</div>
</div>
<div class="section" id="tricky">
<h2>Tricky<a class="headerlink" href="#tricky" title="Permalink to this headline"></a></h2>
<p>Let’s think.</p>
<p>Given a <em>sorted</em> sequence (from highest to lowest) we want to * for
head, tail in sequence * for term in tail: * check if the head % term
== 0 * if so compute head / term and terminate loop * else continue</p>
<div class="section" id="so-we-want-a-loop-i-think">
<h3>So we want a <code class="docutils literal notranslate"><span class="pre">loop</span></code> I think<a class="headerlink" href="#so-we-want-a-loop-i-think" title="Permalink to this headline"></a></h3>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="kc">True</span> <span class="p">[</span><span class="n">Q</span><span class="p">]</span> <span class="n">loop</span>
<span class="p">[</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">Q</span> <span class="p">[</span><span class="n">Q</span><span class="p">]</span> <span class="n">loop</span>
</pre></div>
</div>
<p><code class="docutils literal notranslate"><span class="pre">Q</span></code> should either leave the result and False, or the <code class="docutils literal notranslate"><span class="pre">rest</span></code> and
True.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="p">[</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">Q</span>
<span class="o">-----------------</span>
<span class="n">result</span> <span class="mi">0</span>
<span class="p">[</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">Q</span>
<span class="o">-----------------</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="mi">1</span>
</pre></div>
</div>
<p>This suggests that <code class="docutils literal notranslate"><span class="pre">Q</span></code> should start with:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">uncons</span> <span class="n">dup</span> <span class="n">roll</span><span class="o">&lt;</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span>
</pre></div>
</div>
<p>Now we just have to <code class="docutils literal notranslate"><span class="pre">pop</span></code> it if we don’t need it.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="p">[</span><span class="n">P</span><span class="p">]</span> <span class="p">[</span><span class="n">T</span><span class="p">]</span> <span class="p">[</span><span class="n">cons</span><span class="p">]</span> <span class="n">app2</span> <span class="n">popdd</span> <span class="p">[</span><span class="n">E</span><span class="p">]</span> <span class="n">primrec</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">a</span> <span class="n">P</span><span class="p">]</span> <span class="p">[</span><span class="n">a</span> <span class="n">T</span><span class="p">]</span> <span class="p">[</span><span class="n">E</span><span class="p">]</span> <span class="n">primrec</span>
</pre></div>
</div>
<hr class="docutils" />
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">w</span><span class="o">/</span> <span class="n">Q</span> <span class="o">==</span> <span class="p">[</span><span class="o">%</span> <span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">T</span><span class="p">]</span> <span class="p">[</span><span class="n">F</span><span class="p">]</span> <span class="n">primrec</span>
<span class="p">[</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">uncons</span>
<span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">tuck</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">uncons</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="n">b</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">roll</span><span class="o">&gt;</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="n">b</span> <span class="n">Q</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="n">b</span> <span class="p">[</span><span class="o">%</span> <span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">T</span><span class="p">]</span> <span class="p">[</span><span class="n">F</span><span class="p">]</span> <span class="n">primrec</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="n">b</span> <span class="n">T</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="n">b</span> <span class="o">/</span> <span class="n">roll</span><span class="o">&gt;</span> <span class="n">popop</span> <span class="mi">0</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="n">b</span> <span class="n">F</span> <span class="n">Q</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="n">b</span> <span class="n">pop</span> <span class="n">swap</span> <span class="n">uncons</span> <span class="o">...</span> <span class="n">Q</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="n">swap</span> <span class="n">uncons</span> <span class="o">...</span> <span class="n">Q</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">uncons</span> <span class="o">...</span> <span class="n">Q</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="n">c</span> <span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="n">roll</span><span class="o">&gt;</span> <span class="n">Q</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="n">c</span> <span class="n">Q</span>
<span class="n">Q</span> <span class="o">==</span> <span class="p">[</span><span class="o">%</span> <span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="o">/</span> <span class="n">roll</span><span class="o">&gt;</span> <span class="n">popop</span> <span class="mi">0</span><span class="p">]</span> <span class="p">[</span><span class="n">pop</span> <span class="n">swap</span> <span class="n">uncons</span> <span class="n">roll</span><span class="o">&gt;</span><span class="p">]</span> <span class="n">primrec</span>
<span class="n">uncons</span> <span class="n">tuck</span> <span class="n">uncons</span> <span class="n">roll</span><span class="o">&gt;</span> <span class="n">Q</span>
</pre></div>
</div>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;[8 5 3 2] 9 [swap] [% not] [cons] app2 popdd&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="mi">8</span> <span class="mi">5</span> <span class="mi">3</span> <span class="mi">2</span><span class="p">]</span> <span class="p">[</span><span class="mi">9</span> <span class="n">swap</span><span class="p">]</span> <span class="p">[</span><span class="mi">9</span> <span class="o">%</span> <span class="ow">not</span><span class="p">]</span>
</pre></div>
</div>
<hr class="docutils" />
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="p">[</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">uncons</span>
<span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">tuck</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">1</span><span class="p">]</span> <span class="p">[</span><span class="n">Q</span><span class="p">]</span> <span class="n">ifte</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="p">[]</span> <span class="n">popop</span> <span class="mi">1</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="mi">1</span>
<span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">Q</span>
<span class="n">a</span> <span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="n">Q</span>
<span class="o">---------------</span>
<span class="n">result</span> <span class="mi">0</span>
<span class="n">a</span> <span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="n">Q</span>
<span class="o">---------------</span>
<span class="mi">1</span>
<span class="n">w</span><span class="o">/</span> <span class="n">Q</span> <span class="o">==</span> <span class="p">[</span><span class="n">first</span> <span class="o">%</span> <span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">first</span> <span class="o">/</span> <span class="mi">0</span><span class="p">]</span> <span class="p">[</span><span class="n">rest</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">1</span><span class="p">]]</span> <span class="p">[</span><span class="n">ifte</span><span class="p">]</span>
<span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">first</span> <span class="o">%</span> <span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">first</span> <span class="o">/</span> <span class="mi">0</span><span class="p">]</span> <span class="p">[</span><span class="n">rest</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">1</span><span class="p">]]</span> <span class="p">[</span><span class="n">ifte</span><span class="p">]</span>
<span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">first</span> <span class="o">%</span> <span class="ow">not</span>
<span class="n">a</span> <span class="n">b</span> <span class="o">%</span> <span class="ow">not</span>
<span class="n">a</span><span class="o">%</span><span class="n">b</span> <span class="ow">not</span>
<span class="nb">bool</span><span class="p">(</span><span class="n">a</span><span class="o">%</span><span class="n">b</span><span class="p">)</span>
<span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">first</span> <span class="o">%</span> <span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">first</span> <span class="o">/</span> <span class="mi">0</span><span class="p">]</span> <span class="p">[</span><span class="n">rest</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">1</span><span class="p">]]</span> <span class="p">[</span><span class="n">ifte</span><span class="p">]</span>
<span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">first</span> <span class="o">/</span> <span class="mi">0</span>
<span class="n">a</span> <span class="n">b</span> <span class="o">/</span> <span class="mi">0</span>
<span class="n">a</span><span class="o">/</span><span class="n">b</span> <span class="mi">0</span>
<span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="n">first</span> <span class="o">%</span> <span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">first</span> <span class="o">/</span> <span class="mi">0</span><span class="p">]</span> <span class="p">[</span><span class="n">rest</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">1</span><span class="p">]]</span> <span class="p">[</span><span class="n">ifte</span><span class="p">]</span>
<span class="n">a</span> <span class="p">[</span><span class="n">b</span> <span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">rest</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">1</span><span class="p">]</span> <span class="p">[</span><span class="n">Q</span><span class="p">]</span> <span class="n">ifte</span>
<span class="n">a</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">1</span><span class="p">]</span> <span class="p">[</span><span class="n">Q</span><span class="p">]</span> <span class="n">ifte</span>
<span class="n">a</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">1</span><span class="p">]</span> <span class="p">[</span><span class="n">Q</span><span class="p">]</span> <span class="n">ifte</span>
<span class="n">a</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">1</span><span class="p">]</span> <span class="p">[</span><span class="n">Q</span><span class="p">]</span> <span class="n">ifte</span>
<span class="n">a</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="ow">not</span>
<span class="n">a</span> <span class="p">[]</span> <span class="n">popop</span> <span class="mi">1</span>
<span class="mi">1</span>
<span class="n">a</span> <span class="p">[</span><span class="n">c</span> <span class="n">d</span><span class="p">]</span> <span class="n">Q</span>
<span class="n">uncons</span> <span class="n">tuck</span> <span class="p">[</span><span class="n">first</span> <span class="o">%</span> <span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">first</span> <span class="o">/</span> <span class="mi">0</span><span class="p">]</span> <span class="p">[</span><span class="n">rest</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">1</span><span class="p">]]</span> <span class="p">[</span><span class="n">ifte</span><span class="p">]</span>
</pre></div>
</div>
</div>
<div class="section" id="i-finally-sat-down-with-a-piece-of-paper-and-blocked-it-out">
<h3>I finally sat down with a piece of paper and blocked it out.<a class="headerlink" href="#i-finally-sat-down-with-a-piece-of-paper-and-blocked-it-out" title="Permalink to this headline"></a></h3>
<p>First, I made a function <code class="docutils literal notranslate"><span class="pre">G</span></code> that expects a number and a sequence of
candidates and return the result or zero:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="n">n</span> <span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="n">G</span>
<span class="o">---------------</span>
<span class="n">result</span>
<span class="n">n</span> <span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="n">G</span>
<span class="o">---------------</span>
<span class="mi">0</span>
</pre></div>
</div>
<p>It’s a recursive function that conditionally executes the recursive part
of its recursive branch</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="n">Pg</span><span class="p">]</span> <span class="p">[</span><span class="n">E</span><span class="p">]</span> <span class="p">[</span><span class="n">R1</span> <span class="p">[</span><span class="n">Pi</span><span class="p">]</span> <span class="p">[</span><span class="n">T</span><span class="p">]]</span> <span class="p">[</span><span class="n">ifte</span><span class="p">]</span> <span class="n">genrec</span>
</pre></div>
</div>
<p>The recursive branch is the else-part of the inner <code class="docutils literal notranslate"><span class="pre">ifte</span></code>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">G</span> <span class="o">==</span> <span class="p">[</span><span class="n">Pg</span><span class="p">]</span> <span class="p">[</span><span class="n">E</span><span class="p">]</span> <span class="p">[</span><span class="n">R1</span> <span class="p">[</span><span class="n">Pi</span><span class="p">]</span> <span class="p">[</span><span class="n">T</span><span class="p">]]</span> <span class="p">[</span><span class="n">ifte</span><span class="p">]</span> <span class="n">genrec</span>
<span class="o">==</span> <span class="p">[</span><span class="n">Pg</span><span class="p">]</span> <span class="p">[</span><span class="n">E</span><span class="p">]</span> <span class="p">[</span><span class="n">R1</span> <span class="p">[</span><span class="n">Pi</span><span class="p">]</span> <span class="p">[</span><span class="n">T</span><span class="p">]</span> <span class="p">[</span><span class="n">G</span><span class="p">]</span> <span class="n">ifte</span><span class="p">]</span> <span class="n">ifte</span>
</pre></div>
</div>
<p>But this is in hindsight. Going forward I derived:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">G</span> <span class="o">==</span> <span class="p">[</span><span class="n">first</span> <span class="o">%</span> <span class="ow">not</span><span class="p">]</span>
<span class="p">[</span><span class="n">first</span> <span class="o">/</span><span class="p">]</span>
<span class="p">[</span><span class="n">rest</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">0</span><span class="p">]]</span>
<span class="p">[</span><span class="n">ifte</span><span class="p">]</span> <span class="n">genrec</span>
</pre></div>
</div>
<p>The predicate detects if the <code class="docutils literal notranslate"><span class="pre">n</span></code> can be evenly divided by the
<code class="docutils literal notranslate"><span class="pre">first</span></code> item in the list. If so, the then-part returns the result.
Otherwise, we have:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="p">[</span><span class="n">m</span> <span class="o">...</span><span class="p">]</span> <span class="n">rest</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">0</span><span class="p">]</span> <span class="p">[</span><span class="n">G</span><span class="p">]</span> <span class="n">ifte</span>
<span class="n">n</span> <span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="p">[</span><span class="ow">not</span><span class="p">]</span> <span class="p">[</span><span class="n">popop</span> <span class="mi">0</span><span class="p">]</span> <span class="p">[</span><span class="n">G</span><span class="p">]</span> <span class="n">ifte</span>
</pre></div>
</div>
<p>This <code class="docutils literal notranslate"><span class="pre">ifte</span></code> guards against empty sequences and returns zero in that
case, otherwise it executes <code class="docutils literal notranslate"><span class="pre">G</span></code>.</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;G == [first % not] [first /] [rest [not] [popop 0]] [ifte] genrec&#39;</span><span class="p">)</span>
</pre></div>
</div>
<p>Now we need a word that uses <code class="docutils literal notranslate"><span class="pre">G</span></code> on each (head, tail) pair of a
sequence until it finds a (non-zero) result. It’s going to be designed
to work on a stack that has some candidate <code class="docutils literal notranslate"><span class="pre">n</span></code>, a sequence of possible
divisors, and a result that is zero to signal to continue (a non-zero
value implies that it is the discovered result):</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="n">n</span> <span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="n">p</span> <span class="n">find</span><span class="o">-</span><span class="n">result</span>
<span class="o">---------------------------</span>
<span class="n">result</span>
</pre></div>
</div>
<p>It applies <code class="docutils literal notranslate"><span class="pre">G</span></code> using <code class="docutils literal notranslate"><span class="pre">nullary</span></code> because if it fails with one
candidate it needs the list to get the next one (the list is otherwise
consumed by <code class="docutils literal notranslate"><span class="pre">G</span></code>.)</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">find</span><span class="o">-</span><span class="n">result</span> <span class="o">==</span> <span class="p">[</span><span class="mi">0</span> <span class="o">&gt;</span><span class="p">]</span> <span class="p">[</span><span class="n">roll</span><span class="o">&gt;</span> <span class="n">popop</span><span class="p">]</span> <span class="p">[</span><span class="n">roll</span><span class="o">&lt;</span> <span class="n">popop</span> <span class="n">uncons</span> <span class="p">[</span><span class="n">G</span><span class="p">]</span> <span class="n">nullary</span><span class="p">]</span> <span class="n">primrec</span>
<span class="n">n</span> <span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="n">p</span> <span class="p">[</span><span class="mi">0</span> <span class="o">&gt;</span><span class="p">]</span> <span class="p">[</span><span class="n">roll</span><span class="o">&gt;</span> <span class="n">popop</span><span class="p">]</span> <span class="p">[</span><span class="n">roll</span><span class="o">&lt;</span> <span class="n">popop</span> <span class="n">uncons</span> <span class="p">[</span><span class="n">G</span><span class="p">]</span> <span class="n">nullary</span><span class="p">]</span> <span class="n">primrec</span>
</pre></div>
</div>
<p>The base-case is trivial, return the (non-zero) result. The recursive
branch…</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="n">p</span> <span class="n">roll</span><span class="o">&lt;</span> <span class="n">popop</span> <span class="n">uncons</span> <span class="p">[</span><span class="n">G</span><span class="p">]</span> <span class="n">nullary</span> <span class="n">find</span><span class="o">-</span><span class="n">result</span>
<span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="n">p</span> <span class="n">n</span> <span class="n">popop</span> <span class="n">uncons</span> <span class="p">[</span><span class="n">G</span><span class="p">]</span> <span class="n">nullary</span> <span class="n">find</span><span class="o">-</span><span class="n">result</span>
<span class="p">[</span><span class="o">...</span><span class="p">]</span> <span class="n">uncons</span> <span class="p">[</span><span class="n">G</span><span class="p">]</span> <span class="n">nullary</span> <span class="n">find</span><span class="o">-</span><span class="n">result</span>
<span class="n">m</span> <span class="p">[</span><span class="o">..</span><span class="p">]</span> <span class="p">[</span><span class="n">G</span><span class="p">]</span> <span class="n">nullary</span> <span class="n">find</span><span class="o">-</span><span class="n">result</span>
<span class="n">m</span> <span class="p">[</span><span class="o">..</span><span class="p">]</span> <span class="n">p</span> <span class="n">find</span><span class="o">-</span><span class="n">result</span>
</pre></div>
</div>
<p>The puzzle states that the input is well-formed, meaning that we can
expect a result before the row sequence empties and so do not need to
guard the <code class="docutils literal notranslate"><span class="pre">uncons</span></code>.</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;find-result == [0 &gt;] [roll&gt; popop] [roll&lt; popop uncons [G] nullary] primrec&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;[11 9 8 7 3 2] 0 tuck find-result&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">3.0</span>
</pre></div>
</div>
<p>In order to get the thing started, we need to <code class="docutils literal notranslate"><span class="pre">sort</span></code> the list in
descending order, then prime the <code class="docutils literal notranslate"><span class="pre">find-result</span></code> function with a dummy
candidate value and zero (“continue”) flag.</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;prep-row == sort reverse 0 tuck&#39;</span><span class="p">)</span>
</pre></div>
</div>
<p>Now we can define our program.</p>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;AoC20017.2.extra == [prep-row find-result +] step_zero&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="code ipython2 highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;&#39;&#39;</span>
<span class="s1">[[5 9 2 8]</span>
<span class="s1"> [9 4 7 3]</span>
<span class="s1"> [3 8 6 5]] AoC20017.2.extra</span>
<span class="s1">&#39;&#39;&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="mf">9.0</span>
</pre></div>
</div>
</div>
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<li><a class="reference internal" href="#december-2nd">December 2nd</a></li>
<li><a class="reference internal" href="#tricky">Tricky</a><ul>
<li><a class="reference internal" href="#so-we-want-a-loop-i-think">So we want a <code class="docutils literal notranslate"><span class="pre">loop</span></code> I think</a></li>
<li><a class="reference internal" href="#i-finally-sat-down-with-a-piece-of-paper-and-blocked-it-out">I finally sat down with a piece of paper and blocked it out.</a></li>
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