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<?xml version="1.0" encoding="utf-8" ?>
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<div class="document" id="project-euler">
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<h1 class="title">Project Euler 题解第一页</h1>
<p>用 <a class="reference external" href="http://julialang.org/">Julia</a> 做 <a class="reference external" href="http://projecteuler.net">Project Euler</a> 上的题目简直比作弊还简单 ...</p>
<div class="section" id="id1">
<h1>1</h1>
<p>找出 1000 以下 3 或 5 的倍数的和</p>
<p>不用讲解 - -</p>
<pre class="code julia literal-block">
<span class="n">println</span><span class="p">(</span><span class="n">sum</span><span class="p">(</span><span class="n">filter</span><span class="p">(</span><span class="n">i</span><span class="o">-></span><span class="n">i</span><span class="o">%</span><span class="mi">3</span><span class="o">==</span><span class="mi">0</span> <span class="o">||</span> <span class="n">i</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="p">[</span><span class="mi">3</span><span class="o">:</span><span class="mi">999</span><span class="p">])))</span>
</pre>
</div>
<div class="section" id="id2">
<h1>2</h1>
<p>斐波拉切数列的偶数项和</p>
<p>还是不用讲解 - -</p>
<pre class="code julia literal-block">
<span class="n">a</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">b</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">sum</span> <span class="o">=</span> <span class="mi">2</span>
<span class="k">while</span> <span class="kc">true</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">a</span><span class="o">+</span><span class="n">b</span>
<span class="k">if</span> <span class="n">c</span><span class="o">></span><span class="mi">4000000</span> <span class="n">then</span>
<span class="k">break</span>
<span class="k">end</span>
<span class="k">if</span> <span class="n">c</span><span class="o">%</span><span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span> <span class="n">then</span>
<span class="n">sum</span> <span class="o">=</span> <span class="n">sum</span> <span class="o">+</span> <span class="n">c</span>
<span class="k">end</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">b</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">c</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">sum</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id3">
<h1>3</h1>
<p>600851475143 的最大质数因子</p>
<p>对于 julia 这么一种 IMBA 的语言 …… 还是不用讲解</p>
<pre class="code julia literal-block">
<span class="n">v</span> <span class="o">=</span> <span class="mi">600851475143</span>
<span class="n">println</span><span class="p">(</span><span class="n">maximum</span><span class="p">([</span><span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">f</span> <span class="kp">in</span> <span class="n">factor</span><span class="p">(</span><span class="n">v</span><span class="p">)]))</span>
</pre>
</div>
<div class="section" id="id4">
<h1>4</h1>
<p>找出三位数相乘组成的最大回文数</p>
<p>依然不用讲解</p>
<pre class="code julia literal-block">
<span class="n">isPalindrome</span><span class="p">(</span><span class="n">s</span><span class="o">::</span><span class="n">String</span><span class="p">)</span> <span class="o">=</span>
<span class="k">if</span> <span class="n">length</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">></span><span class="mi">2</span>
<span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="n">s</span><span class="p">[</span><span class="k">end</span><span class="p">]</span> <span class="o">&&</span> <span class="n">isPalindrome</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">2</span><span class="o">:</span><span class="k">end</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="k">else</span>
<span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">s</span><span class="p">[</span><span class="k">end</span><span class="p">]</span>
<span class="k">end</span>
<span class="n">isPalindrome</span><span class="p">(</span><span class="n">i</span><span class="o">::</span><span class="kt">Int</span><span class="p">)</span> <span class="o">=</span> <span class="n">isPalindrome</span><span class="p">(</span><span class="n">string</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
<span class="n">println</span><span class="p">(</span><span class="n">maximum</span><span class="p">(</span><span class="n">filter</span><span class="p">(</span><span class="n">isPalindrome</span><span class="p">,</span> <span class="p">[</span><span class="n">a</span><span class="o">*</span><span class="n">b</span> <span class="k">for</span> <span class="n">a</span> <span class="kp">in</span> <span class="mi">100</span><span class="o">:</span><span class="mi">999</span><span class="p">,</span> <span class="n">b</span> <span class="kp">in</span> <span class="mi">100</span><span class="o">:</span><span class="mi">999</span><span class="p">])))</span>
</pre>
</div>
<div class="section" id="id5">
<h1>5</h1>
<p>1 .. 20 的最小公倍数</p>
<pre class="code julia literal-block">
<span class="n">println</span><span class="p">(</span><span class="n">lcm</span><span class="p">([</span><span class="mi">1</span><span class="o">:</span><span class="mi">20</span><span class="p">]</span><span class="o">...</span><span class="p">))</span>
</pre>
</div>
<div class="section" id="id6">
<h1>6</h1>
<p>1 .. 100 和的平方与平方和的差</p>
<pre class="code julia literal-block">
<span class="n">println</span><span class="p">(</span><span class="n">sum</span><span class="p">([</span><span class="mi">1</span><span class="o">:</span><span class="mi">100</span><span class="p">])</span><span class="o">^</span><span class="mi">2</span> <span class="o">-</span> <span class="n">sum</span><span class="p">(</span><span class="n">map</span><span class="p">(</span><span class="n">x</span><span class="o">-></span><span class="n">x</span><span class="o">^</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">100</span><span class="p">])))</span>
</pre>
</div>
<div class="section" id="id7">
<h1>7</h1>
<p>第 10001 个质数是什么呢</p>
<pre class="code julia literal-block">
<span class="n">println</span><span class="p">(</span><span class="n">primes</span><span class="p">(</span><span class="mi">500000</span><span class="p">)[</span><span class="mi">10001</span><span class="p">])</span>
</pre>
</div>
<div class="section" id="id8">
<h1>8</h1>
<p>在那一串数字中找到最大的五个连续的数字的积 ……</p>
<pre class="code julia literal-block">
<span class="n">s</span> <span class="o">=</span> <span class="n">replace</span><span class="p">(</span>
<span class="s">"
73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450
"</span>
<span class="p">,</span> <span class="sr">r"\s"</span><span class="p">,</span> <span class="s">""</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">5</span><span class="o">:</span><span class="n">length</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">max</span><span class="p">(</span> <span class="n">v</span><span class="p">,</span> <span class="n">reduce</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span><span class="o">-></span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">c</span><span class="o">-</span><span class="sc">'0'</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="mi">4</span><span class="o">:</span><span class="n">i</span><span class="p">])</span> <span class="p">)</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id9">
<h1>9</h1>
<p>找到满足 <span class="formula"><i>a</i><sup>2</sup> + <i>b</i><sup>2</sup> = <i>c</i><sup>2</sup></span> 且 <span class="formula"><i>a</i> + <i>b</i> + <i>c</i> = 1000</span> 的 <span class="formula"><i>a</i>, <i>b</i>, <i>c</i></span></p>
<pre class="code julia literal-block">
<span class="k">for</span> <span class="n">a</span> <span class="kp">in</span> <span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">999</span><span class="p">]</span>
<span class="k">for</span> <span class="n">b</span> <span class="kp">in</span> <span class="p">[</span><span class="n">a</span><span class="o">:</span><span class="mi">999</span><span class="p">]</span>
<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="mi">1000</span>
<span class="k">continue</span>
<span class="k">end</span>
<span class="k">for</span> <span class="n">c</span> <span class="kp">in</span> <span class="p">[</span><span class="n">b</span><span class="o">:</span><span class="mi">999</span><span class="p">]</span>
<span class="n">s</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">c</span>
<span class="k">if</span> <span class="n">s</span> <span class="o">></span> <span class="mi">1000</span>
<span class="k">break</span>
<span class="k">end</span>
<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="o">&&</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="mi">1000</span>
<span class="n">println</span><span class="p">(</span><span class="s">"</span><span class="si">$a</span><span class="s"> * </span><span class="si">$b</span><span class="s"> * </span><span class="si">$c</span><span class="s"> = </span><span class="si">$</span><span class="p">(</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="p">)</span><span class="s">"</span><span class="p">)</span>
<span class="n">exit</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">end</span>
</pre>
</div>
<div class="section" id="id10">
<h1>10</h1>
<p>求 2000000 以下的质数的和</p>
<pre class="code julia literal-block">
<span class="n">println</span><span class="p">(</span><span class="n">sum</span><span class="p">(</span><span class="n">primes</span><span class="p">(</span><span class="mi">2000000</span><span class="o">-</span><span class="mi">1</span><span class="p">)))</span>
</pre>
</div>
<div class="section" id="id11">
<h1>11</h1>
<p>求一个矩阵中沿某一方向连续的四个数的最大乘积</p>
<pre class="code julia literal-block">
<span class="n">m</span> <span class="o">=</span> <span class="p">[</span>
<span class="mi">08</span> <span class="mi">02</span> <span class="mi">22</span> <span class="mi">97</span> <span class="mi">38</span> <span class="mi">15</span> <span class="mi">00</span> <span class="mi">40</span> <span class="mi">00</span> <span class="mi">75</span> <span class="mi">04</span> <span class="mi">05</span> <span class="mi">07</span> <span class="mi">78</span> <span class="mi">52</span> <span class="mi">12</span> <span class="mi">50</span> <span class="mi">77</span> <span class="mi">91</span> <span class="mi">08</span>
<span class="mi">49</span> <span class="mi">49</span> <span class="mi">99</span> <span class="mi">40</span> <span class="mi">17</span> <span class="mi">81</span> <span class="mi">18</span> <span class="mi">57</span> <span class="mi">60</span> <span class="mi">87</span> <span class="mi">17</span> <span class="mi">40</span> <span class="mi">98</span> <span class="mi">43</span> <span class="mi">69</span> <span class="mi">48</span> <span class="mi">04</span> <span class="mi">56</span> <span class="mi">62</span> <span class="mi">00</span>
<span class="mi">81</span> <span class="mi">49</span> <span class="mi">31</span> <span class="mi">73</span> <span class="mi">55</span> <span class="mi">79</span> <span class="mi">14</span> <span class="mi">29</span> <span class="mi">93</span> <span class="mi">71</span> <span class="mi">40</span> <span class="mi">67</span> <span class="mi">53</span> <span class="mi">88</span> <span class="mi">30</span> <span class="mi">03</span> <span class="mi">49</span> <span class="mi">13</span> <span class="mi">36</span> <span class="mi">65</span>
<span class="mi">52</span> <span class="mi">70</span> <span class="mi">95</span> <span class="mi">23</span> <span class="mi">04</span> <span class="mi">60</span> <span class="mi">11</span> <span class="mi">42</span> <span class="mi">69</span> <span class="mi">24</span> <span class="mi">68</span> <span class="mi">56</span> <span class="mi">01</span> <span class="mi">32</span> <span class="mi">56</span> <span class="mi">71</span> <span class="mi">37</span> <span class="mi">02</span> <span class="mi">36</span> <span class="mi">91</span>
<span class="mi">22</span> <span class="mi">31</span> <span class="mi">16</span> <span class="mi">71</span> <span class="mi">51</span> <span class="mi">67</span> <span class="mi">63</span> <span class="mi">89</span> <span class="mi">41</span> <span class="mi">92</span> <span class="mi">36</span> <span class="mi">54</span> <span class="mi">22</span> <span class="mi">40</span> <span class="mi">40</span> <span class="mi">28</span> <span class="mi">66</span> <span class="mi">33</span> <span class="mi">13</span> <span class="mi">80</span>
<span class="mi">24</span> <span class="mi">47</span> <span class="mi">32</span> <span class="mi">60</span> <span class="mi">99</span> <span class="mi">03</span> <span class="mi">45</span> <span class="mi">02</span> <span class="mi">44</span> <span class="mi">75</span> <span class="mi">33</span> <span class="mi">53</span> <span class="mi">78</span> <span class="mi">36</span> <span class="mi">84</span> <span class="mi">20</span> <span class="mi">35</span> <span class="mi">17</span> <span class="mi">12</span> <span class="mi">50</span>
<span class="mi">32</span> <span class="mi">98</span> <span class="mi">81</span> <span class="mi">28</span> <span class="mi">64</span> <span class="mi">23</span> <span class="mi">67</span> <span class="mi">10</span> <span class="mi">26</span> <span class="mi">38</span> <span class="mi">40</span> <span class="mi">67</span> <span class="mi">59</span> <span class="mi">54</span> <span class="mi">70</span> <span class="mi">66</span> <span class="mi">18</span> <span class="mi">38</span> <span class="mi">64</span> <span class="mi">70</span>
<span class="mi">67</span> <span class="mi">26</span> <span class="mi">20</span> <span class="mi">68</span> <span class="mi">02</span> <span class="mi">62</span> <span class="mi">12</span> <span class="mi">20</span> <span class="mi">95</span> <span class="mi">63</span> <span class="mi">94</span> <span class="mi">39</span> <span class="mi">63</span> <span class="mi">08</span> <span class="mi">40</span> <span class="mi">91</span> <span class="mi">66</span> <span class="mi">49</span> <span class="mi">94</span> <span class="mi">21</span>
<span class="mi">24</span> <span class="mi">55</span> <span class="mi">58</span> <span class="mi">05</span> <span class="mi">66</span> <span class="mi">73</span> <span class="mi">99</span> <span class="mi">26</span> <span class="mi">97</span> <span class="mi">17</span> <span class="mi">78</span> <span class="mi">78</span> <span class="mi">96</span> <span class="mi">83</span> <span class="mi">14</span> <span class="mi">88</span> <span class="mi">34</span> <span class="mi">89</span> <span class="mi">63</span> <span class="mi">72</span>
<span class="mi">21</span> <span class="mi">36</span> <span class="mi">23</span> <span class="mi">09</span> <span class="mi">75</span> <span class="mi">00</span> <span class="mi">76</span> <span class="mi">44</span> <span class="mi">20</span> <span class="mi">45</span> <span class="mi">35</span> <span class="mi">14</span> <span class="mi">00</span> <span class="mi">61</span> <span class="mi">33</span> <span class="mi">97</span> <span class="mi">34</span> <span class="mi">31</span> <span class="mi">33</span> <span class="mi">95</span>
<span class="mi">78</span> <span class="mi">17</span> <span class="mi">53</span> <span class="mi">28</span> <span class="mi">22</span> <span class="mi">75</span> <span class="mi">31</span> <span class="mi">67</span> <span class="mi">15</span> <span class="mi">94</span> <span class="mi">03</span> <span class="mi">80</span> <span class="mi">04</span> <span class="mi">62</span> <span class="mi">16</span> <span class="mi">14</span> <span class="mi">09</span> <span class="mi">53</span> <span class="mi">56</span> <span class="mi">92</span>
<span class="mi">16</span> <span class="mi">39</span> <span class="mi">05</span> <span class="mi">42</span> <span class="mi">96</span> <span class="mi">35</span> <span class="mi">31</span> <span class="mi">47</span> <span class="mi">55</span> <span class="mi">58</span> <span class="mi">88</span> <span class="mi">24</span> <span class="mi">00</span> <span class="mi">17</span> <span class="mi">54</span> <span class="mi">24</span> <span class="mi">36</span> <span class="mi">29</span> <span class="mi">85</span> <span class="mi">57</span>
<span class="mi">86</span> <span class="mi">56</span> <span class="mi">00</span> <span class="mi">48</span> <span class="mi">35</span> <span class="mi">71</span> <span class="mi">89</span> <span class="mi">07</span> <span class="mi">05</span> <span class="mi">44</span> <span class="mi">44</span> <span class="mi">37</span> <span class="mi">44</span> <span class="mi">60</span> <span class="mi">21</span> <span class="mi">58</span> <span class="mi">51</span> <span class="mi">54</span> <span class="mi">17</span> <span class="mi">58</span>
<span class="mi">19</span> <span class="mi">80</span> <span class="mi">81</span> <span class="mi">68</span> <span class="mi">05</span> <span class="mi">94</span> <span class="mi">47</span> <span class="mi">69</span> <span class="mi">28</span> <span class="mi">73</span> <span class="mi">92</span> <span class="mi">13</span> <span class="mi">86</span> <span class="mi">52</span> <span class="mi">17</span> <span class="mi">77</span> <span class="mi">04</span> <span class="mi">89</span> <span class="mi">55</span> <span class="mi">40</span>
<span class="mi">04</span> <span class="mi">52</span> <span class="mi">08</span> <span class="mi">83</span> <span class="mi">97</span> <span class="mi">35</span> <span class="mi">99</span> <span class="mi">16</span> <span class="mi">07</span> <span class="mi">97</span> <span class="mi">57</span> <span class="mi">32</span> <span class="mi">16</span> <span class="mi">26</span> <span class="mi">26</span> <span class="mi">79</span> <span class="mi">33</span> <span class="mi">27</span> <span class="mi">98</span> <span class="mi">66</span>
<span class="mi">88</span> <span class="mi">36</span> <span class="mi">68</span> <span class="mi">87</span> <span class="mi">57</span> <span class="mi">62</span> <span class="mi">20</span> <span class="mi">72</span> <span class="mi">03</span> <span class="mi">46</span> <span class="mi">33</span> <span class="mi">67</span> <span class="mi">46</span> <span class="mi">55</span> <span class="mi">12</span> <span class="mi">32</span> <span class="mi">63</span> <span class="mi">93</span> <span class="mi">53</span> <span class="mi">69</span>
<span class="mi">04</span> <span class="mi">42</span> <span class="mi">16</span> <span class="mi">73</span> <span class="mi">38</span> <span class="mi">25</span> <span class="mi">39</span> <span class="mi">11</span> <span class="mi">24</span> <span class="mi">94</span> <span class="mi">72</span> <span class="mi">18</span> <span class="mi">08</span> <span class="mi">46</span> <span class="mi">29</span> <span class="mi">32</span> <span class="mi">40</span> <span class="mi">62</span> <span class="mi">76</span> <span class="mi">36</span>
<span class="mi">20</span> <span class="mi">69</span> <span class="mi">36</span> <span class="mi">41</span> <span class="mi">72</span> <span class="mi">30</span> <span class="mi">23</span> <span class="mi">88</span> <span class="mi">34</span> <span class="mi">62</span> <span class="mi">99</span> <span class="mi">69</span> <span class="mi">82</span> <span class="mi">67</span> <span class="mi">59</span> <span class="mi">85</span> <span class="mi">74</span> <span class="mi">04</span> <span class="mi">36</span> <span class="mi">16</span>
<span class="mi">20</span> <span class="mi">73</span> <span class="mi">35</span> <span class="mi">29</span> <span class="mi">78</span> <span class="mi">31</span> <span class="mi">90</span> <span class="mi">01</span> <span class="mi">74</span> <span class="mi">31</span> <span class="mi">49</span> <span class="mi">71</span> <span class="mi">48</span> <span class="mi">86</span> <span class="mi">81</span> <span class="mi">16</span> <span class="mi">23</span> <span class="mi">57</span> <span class="mi">05</span> <span class="mi">54</span>
<span class="mi">01</span> <span class="mi">70</span> <span class="mi">54</span> <span class="mi">71</span> <span class="mi">83</span> <span class="mi">51</span> <span class="mi">54</span> <span class="mi">69</span> <span class="mi">16</span> <span class="mi">92</span> <span class="mi">33</span> <span class="mi">48</span> <span class="mi">61</span> <span class="mi">43</span> <span class="mi">52</span> <span class="mi">01</span> <span class="mi">89</span> <span class="mi">19</span> <span class="mi">67</span> <span class="mi">48</span>
<span class="p">]</span>
<span class="n">d</span> <span class="o">=</span> <span class="mi">20</span> <span class="c"># dimension</span>
<span class="n">v</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">r</span> <span class="kp">in</span> <span class="p">[</span><span class="mi">4</span><span class="o">:</span><span class="n">d</span><span class="p">]</span>
<span class="k">for</span> <span class="n">c</span> <span class="kp">in</span> <span class="p">[</span><span class="mi">4</span><span class="o">:</span><span class="n">d</span><span class="p">]</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">max</span><span class="p">(</span><span class="n">v</span><span class="p">,</span>
<span class="n">prod</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="n">r</span> <span class="p">,</span> <span class="n">c</span><span class="o">-</span><span class="mi">3</span><span class="o">:</span><span class="n">c</span><span class="p">]),</span>
<span class="n">prod</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="n">r</span><span class="o">-</span><span class="mi">3</span><span class="o">:</span><span class="n">r</span> <span class="p">,</span> <span class="n">c</span><span class="p">]),</span>
<span class="n">m</span><span class="p">[</span><span class="n">r</span> <span class="p">,</span> <span class="n">c</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span><span class="p">[</span><span class="n">r</span><span class="o">-</span><span class="mi">1</span> <span class="p">,</span> <span class="n">c</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span><span class="p">[</span><span class="n">r</span><span class="o">-</span><span class="mi">2</span> <span class="p">,</span> <span class="n">c</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span><span class="p">[</span><span class="n">r</span><span class="o">-</span><span class="mi">3</span> <span class="p">,</span> <span class="n">c</span><span class="o">-</span><span class="mi">3</span><span class="p">],</span>
<span class="n">m</span><span class="p">[</span><span class="n">r</span> <span class="p">,</span> <span class="n">c</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span><span class="p">[</span><span class="n">r</span><span class="o">-</span><span class="mi">1</span> <span class="p">,</span> <span class="n">c</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span><span class="p">[</span><span class="n">r</span><span class="o">-</span><span class="mi">2</span> <span class="p">,</span> <span class="n">c</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span><span class="p">[</span><span class="n">r</span><span class="o">-</span><span class="mi">3</span> <span class="p">,</span> <span class="n">c</span><span class="p">]</span>
<span class="p">)</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id12">
<h1>12</h1>
<p>求因数的数量 // 即质因数的组合方式个数</p>
<p>精髓在这一句 <tt class="docutils literal"><span class="pre">reduce((a,</span> <span class="pre">f)->a</span> * <span class="pre">(f[2]+1),</span> 1, factor(t))</tt></p>
<p><tt class="docutils literal">factor(x)</tt> 返回 <tt class="docutils literal">x</tt> 的质因数及其次数列表</p>
<pre class="code julia literal-block">
<span class="n">t</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">100000</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">t</span><span class="o">+</span><span class="n">i</span>
<span class="k">if</span> <span class="n">reduce</span><span class="p">((</span><span class="n">a</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span><span class="o">-></span><span class="n">a</span> <span class="o">*</span> <span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">factor</span><span class="p">(</span><span class="n">t</span><span class="p">))</span><span class="o">></span><span class="mi">500</span>
<span class="n">println</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="k">break</span>
<span class="k">end</span>
<span class="k">end</span>
</pre>
</div>
<div class="section" id="id13">
<h1>13</h1>
<p>求一串大数的和</p>
<pre class="code julia literal-block">
<span class="n">nums</span> <span class="o">=</span> <span class="n">map</span><span class="p">(</span><span class="kt">BigInt</span><span class="p">,[</span>
<span class="s">"37107287533902102798797998220837590246510135740250"</span>
<span class="o">...</span>
<span class="s">"53503534226472524250874054075591789781264330331690"</span>
<span class="p">])</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">string</span><span class="p">(</span><span class="n">sum</span><span class="p">(</span><span class="n">nums</span><span class="p">))</span>
<span class="n">println</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">10</span><span class="p">])</span>
</pre>
</div>
<div class="section" id="id14">
<h1>14</h1>
<p>求使得 3n+1 数列最长的起点</p>
<p>——本该用 DP 做的 …… 不过既然直接算时间也能接受 …… 那就不管了</p>
<pre class="code julia literal-block">
<span class="k">function</span> <span class="n">chainlen</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="k">function</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">if</span> <span class="n">x</span><span class="o">%</span><span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span>
<span class="k">return</span> <span class="n">x</span><span class="o">/</span><span class="mi">2</span>
<span class="k">else</span>
<span class="k">return</span> <span class="mi">3</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">l</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="kc">true</span>
<span class="n">l</span> <span class="o">=</span> <span class="n">l</span><span class="o">+</span><span class="mi">1</span>
<span class="k">if</span> <span class="n">n</span><span class="o">==</span><span class="mi">1</span>
<span class="k">break</span>
<span class="k">end</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">l</span>
<span class="k">end</span>
<span class="n">n</span><span class="p">,</span> <span class="n">l</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">3</span><span class="o">:</span><span class="mi">1000000</span><span class="o">-</span><span class="mi">1</span>
<span class="n">cl</span> <span class="o">=</span> <span class="n">chainlen</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="k">if</span> <span class="n">cl</span><span class="o">></span><span class="n">l</span>
<span class="n">n</span><span class="p">,</span> <span class="n">l</span> <span class="o">=</span> <span class="n">i</span><span class="p">,</span> <span class="n">cl</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id15">
<h1>15</h1>
<p>求在 20x20 的格子上从一个角走到对角不走回头路的路径个数</p>
<p>从一行 n 列开始递推 n 行 n 列的情况就很明确了</p>
<p>需要留意的是 20x20 的格子有 21x21 行列 - -|||</p>
<pre class="code julia literal-block">
<span class="n">row</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">21</span><span class="p">]</span>
<span class="n">nextrow</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">21</span><span class="p">]</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">20</span>
<span class="k">for</span> <span class="n">j</span> <span class="kp">in</span> <span class="mi">2</span><span class="o">:</span><span class="mi">21</span>
<span class="n">nextrow</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">nextrow</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">row</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
<span class="k">end</span>
<span class="n">row</span><span class="p">,</span> <span class="n">nextrow</span> <span class="o">=</span> <span class="n">nextrow</span><span class="p">,</span> <span class="n">row</span>
<span class="c"># println(row)</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span> <span class="n">row</span><span class="p">[</span><span class="mi">21</span><span class="p">]</span> <span class="p">)</span>
</pre>
</div>
<div class="section" id="id16">
<h1>16</h1>
<p>求 <span class="formula">2<sup>1000</sup></span> 的每位数字的和</p>
<pre class="code julia literal-block">
<span class="n">println</span><span class="p">(</span><span class="n">reduce</span><span class="p">((</span><span class="n">a</span><span class="p">,</span><span class="n">c</span><span class="p">)</span><span class="o">-></span><span class="n">a</span><span class="o">+</span><span class="p">(</span><span class="n">c</span><span class="o">-</span><span class="sc">'0'</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">string</span><span class="p">(</span><span class="kt">BigInt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">^</span><span class="mi">1000</span><span class="p">)))</span>
</pre>
</div>
<div class="section" id="id17">
<h1>17</h1>
<p>求用英语写出 1 .. 1000 每一个数字，一共有多少个字母</p>
<pre class="code julia literal-block">
<span class="n">dic</span> <span class="o">=</span> <span class="p">[</span>
<span class="mi">1</span><span class="o">=></span><span class="s">"one"</span><span class="p">,</span>
<span class="mi">2</span><span class="o">=></span><span class="s">"two"</span><span class="p">,</span>
<span class="mi">3</span><span class="o">=></span><span class="s">"three"</span><span class="p">,</span>
<span class="mi">4</span><span class="o">=></span><span class="s">"four"</span><span class="p">,</span>
<span class="mi">5</span><span class="o">=></span><span class="s">"five"</span><span class="p">,</span>
<span class="mi">6</span><span class="o">=></span><span class="s">"six"</span><span class="p">,</span>
<span class="mi">7</span><span class="o">=></span><span class="s">"seven"</span><span class="p">,</span>
<span class="mi">8</span><span class="o">=></span><span class="s">"eight"</span><span class="p">,</span>
<span class="mi">9</span><span class="o">=></span><span class="s">"nine"</span><span class="p">,</span>
<span class="mi">10</span><span class="o">=></span><span class="s">"ten"</span><span class="p">,</span>
<span class="mi">11</span><span class="o">=></span><span class="s">"eleven"</span><span class="p">,</span>
<span class="mi">12</span><span class="o">=></span><span class="s">"twelve"</span><span class="p">,</span>
<span class="mi">13</span><span class="o">=></span><span class="s">"thirteen"</span><span class="p">,</span>
<span class="mi">14</span><span class="o">=></span><span class="s">"fourteen"</span><span class="p">,</span>
<span class="mi">15</span><span class="o">=></span><span class="s">"fifteen"</span><span class="p">,</span>
<span class="mi">16</span><span class="o">=></span><span class="s">"sixteen"</span><span class="p">,</span>
<span class="mi">17</span><span class="o">=></span><span class="s">"seventeen"</span><span class="p">,</span>
<span class="mi">18</span><span class="o">=></span><span class="s">"eighteen"</span><span class="p">,</span>
<span class="mi">19</span><span class="o">=></span><span class="s">"nineteen"</span><span class="p">,</span>
<span class="mi">20</span><span class="o">=></span><span class="s">"twenty"</span><span class="p">,</span>
<span class="mi">30</span><span class="o">=></span><span class="s">"thirty"</span><span class="p">,</span>
<span class="mi">40</span><span class="o">=></span><span class="s">"forty"</span><span class="p">,</span>
<span class="mi">50</span><span class="o">=></span><span class="s">"fifty"</span><span class="p">,</span>
<span class="mi">60</span><span class="o">=></span><span class="s">"sixty"</span><span class="p">,</span>
<span class="mi">70</span><span class="o">=></span><span class="s">"seventy"</span><span class="p">,</span>
<span class="mi">80</span><span class="o">=></span><span class="s">"eighty"</span><span class="p">,</span>
<span class="mi">90</span><span class="o">=></span><span class="s">"ninety"</span><span class="p">,</span>
<span class="mi">100</span><span class="o">=></span><span class="s">"hundred"</span><span class="p">,</span>
<span class="mi">1000</span><span class="o">=></span><span class="s">"thousand"</span><span class="p">,</span>
<span class="p">]</span>
<span class="o">+</span><span class="p">(</span><span class="n">s1</span><span class="o">::</span><span class="n">String</span><span class="p">,</span> <span class="n">s2</span><span class="o">::</span><span class="n">String</span><span class="p">)</span> <span class="o">=</span> <span class="s">"</span><span class="si">$s1$s2</span><span class="s">"</span>
<span class="k">function</span> <span class="n">toeng</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">assert</span><span class="p">(</span><span class="n">n</span><span class="o">></span><span class="mi">0</span> <span class="o">&&</span> <span class="n">n</span><span class="o"><=</span><span class="mi">1000</span><span class="p">)</span>
<span class="k">if</span> <span class="n">n</span><span class="o"><=</span><span class="mi">20</span>
<span class="k">return</span> <span class="n">dic</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
<span class="k">end</span>
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1000</span>
<span class="k">return</span> <span class="s">"one thousand"</span>
<span class="k">end</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">int</span><span class="p">(</span><span class="n">floor</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="mi">100</span><span class="p">))</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">int</span><span class="p">(</span><span class="n">floor</span><span class="p">((</span><span class="n">n</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="mi">100</span><span class="p">)</span><span class="o">/</span><span class="mi">10</span><span class="p">))</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">int</span><span class="p">(</span><span class="n">floor</span><span class="p">((</span><span class="n">n</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="mi">100</span><span class="o">-</span><span class="n">b</span><span class="o">*</span><span class="mi">10</span><span class="p">)))</span>
<span class="n">en</span> <span class="o">=</span> <span class="s">""</span>
<span class="k">if</span> <span class="n">a</span> <span class="o">></span> <span class="mi">0</span>
<span class="n">en</span> <span class="o">=</span> <span class="n">dic</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">+</span> <span class="s">" "</span> <span class="o">+</span> <span class="n">dic</span><span class="p">[</span><span class="mi">100</span><span class="p">]</span>
<span class="k">if</span> <span class="n">b</span> <span class="o">></span> <span class="mi">0</span> <span class="o">||</span> <span class="n">c</span> <span class="o">></span> <span class="mi">0</span>
<span class="n">en</span> <span class="o">=</span> <span class="n">en</span> <span class="o">+</span> <span class="s">" and "</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="mi">1</span> <span class="o">||</span> <span class="n">b</span> <span class="o">==</span> <span class="mi">0</span> <span class="o">&&</span> <span class="n">c</span> <span class="o">></span> <span class="mi">0</span>
<span class="n">en</span> <span class="o">=</span> <span class="n">en</span> <span class="o">+</span> <span class="n">dic</span><span class="p">[</span><span class="n">b</span><span class="o">*</span><span class="mi">10</span><span class="o">+</span><span class="n">c</span><span class="p">]</span>
<span class="k">elseif</span> <span class="n">b</span> <span class="o">></span> <span class="mi">1</span>
<span class="n">en</span> <span class="o">=</span> <span class="n">en</span> <span class="o">+</span> <span class="n">dic</span><span class="p">[</span><span class="n">b</span><span class="o">*</span><span class="mi">10</span><span class="p">]</span>
<span class="k">if</span> <span class="n">c</span> <span class="o">></span> <span class="mi">0</span>
<span class="n">en</span> <span class="o">=</span> <span class="n">en</span> <span class="o">+</span> <span class="s">"-"</span> <span class="o">+</span> <span class="n">dic</span><span class="p">[</span><span class="n">c</span><span class="p">]</span>
<span class="k">end</span>
<span class="k">elseif</span> <span class="n">c</span><span class="o">></span><span class="mi">0</span>
<span class="n">en</span> <span class="o">=</span> <span class="n">en</span> <span class="o">+</span> <span class="n">dic</span><span class="p">[</span><span class="n">c</span><span class="p">]</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">en</span>
<span class="k">end</span>
<span class="c"># println(toeng(342))</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">1000</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">reduce</span><span class="p">(</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span><span class="o">-></span><span class="k">if</span> <span class="n">c</span><span class="o">>=</span><span class="sc">'a'</span> <span class="o">&&</span> <span class="n">c</span><span class="o"><=</span><span class="sc">'z'</span> <span class="n">n</span><span class="o">+</span><span class="mi">1</span> <span class="k">else</span> <span class="n">n</span> <span class="k">end</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">toeng</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="p">)</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id18">
<h1>18</h1>
<p>求从数字三角形顶部走到底，路过的每一位数字相加所能得到的最大值</p>
<p>从底部向上递推就好了</p>
<pre class="code julia literal-block">
<span class="n">m</span> <span class="o">=</span> <span class="p">[</span>
<span class="mi">75</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">95</span> <span class="mi">64</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">17</span> <span class="mi">47</span> <span class="mi">82</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">18</span> <span class="mi">35</span> <span class="mi">87</span> <span class="mi">10</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">20</span> <span class="mi">04</span> <span class="mi">82</span> <span class="mi">47</span> <span class="mi">65</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">19</span> <span class="mi">01</span> <span class="mi">23</span> <span class="mi">75</span> <span class="mi">03</span> <span class="mi">34</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">88</span> <span class="mi">02</span> <span class="mi">77</span> <span class="mi">73</span> <span class="mi">07</span> <span class="mi">63</span> <span class="mi">67</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">99</span> <span class="mi">65</span> <span class="mi">04</span> <span class="mi">28</span> <span class="mi">06</span> <span class="mi">16</span> <span class="mi">70</span> <span class="mi">92</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">41</span> <span class="mi">41</span> <span class="mi">26</span> <span class="mi">56</span> <span class="mi">83</span> <span class="mi">40</span> <span class="mi">80</span> <span class="mi">70</span> <span class="mi">33</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">41</span> <span class="mi">48</span> <span class="mi">72</span> <span class="mi">33</span> <span class="mi">47</span> <span class="mi">32</span> <span class="mi">37</span> <span class="mi">16</span> <span class="mi">94</span> <span class="mi">29</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">53</span> <span class="mi">71</span> <span class="mi">44</span> <span class="mi">65</span> <span class="mi">25</span> <span class="mi">43</span> <span class="mi">91</span> <span class="mi">52</span> <span class="mi">97</span> <span class="mi">51</span> <span class="mi">14</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">70</span> <span class="mi">11</span> <span class="mi">33</span> <span class="mi">28</span> <span class="mi">77</span> <span class="mi">73</span> <span class="mi">17</span> <span class="mi">78</span> <span class="mi">39</span> <span class="mi">68</span> <span class="mi">17</span> <span class="mi">57</span> <span class="mi">00</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">91</span> <span class="mi">71</span> <span class="mi">52</span> <span class="mi">38</span> <span class="mi">17</span> <span class="mi">14</span> <span class="mi">91</span> <span class="mi">43</span> <span class="mi">58</span> <span class="mi">50</span> <span class="mi">27</span> <span class="mi">29</span> <span class="mi">48</span> <span class="mi">00</span> <span class="mi">00</span>
<span class="mi">63</span> <span class="mi">66</span> <span class="mi">04</span> <span class="mi">68</span> <span class="mi">89</span> <span class="mi">53</span> <span class="mi">67</span> <span class="mi">30</span> <span class="mi">73</span> <span class="mi">16</span> <span class="mi">69</span> <span class="mi">87</span> <span class="mi">40</span> <span class="mi">31</span> <span class="mi">00</span>
<span class="mi">04</span> <span class="mi">62</span> <span class="mi">98</span> <span class="mi">27</span> <span class="mi">23</span> <span class="mi">09</span> <span class="mi">70</span> <span class="mi">98</span> <span class="mi">73</span> <span class="mi">93</span> <span class="mi">38</span> <span class="mi">53</span> <span class="mi">60</span> <span class="mi">04</span> <span class="mi">23</span>
<span class="p">]</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">14</span><span class="o">:-</span><span class="mi">1</span><span class="o">:</span><span class="mi">1</span>
<span class="k">for</span> <span class="n">j</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="n">i</span>
<span class="n">m</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">max</span><span class="p">(</span><span class="n">m</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="n">j</span><span class="p">],</span> <span class="n">m</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="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
</pre>
</div>
<div class="section" id="id19">
<h1>19</h1>
<p>求 1901年1月1日 到 2000年12月30日 之间有几个星期天是在月初的</p>
<p>首先呢，Pkg.add("Datetime") // 这真的是作弊好吗！</p>
<pre class="code julia literal-block">
<span class="k">using</span> <span class="n">Datetime</span>
<span class="n">println</span><span class="p">(</span><span class="n">length</span><span class="p">(</span><span class="n">filter</span><span class="p">(</span><span class="n">d</span><span class="o">-></span><span class="n">dayofweek</span><span class="p">(</span><span class="n">d</span><span class="p">)</span><span class="o">==</span><span class="mi">0</span><span class="p">,</span> <span class="n">date</span><span class="p">(</span><span class="mi">1901</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">months</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">:</span><span class="n">date</span><span class="p">(</span><span class="mi">2000</span><span class="p">,</span><span class="mi">12</span><span class="p">,</span><span class="mi">31</span><span class="p">))))</span>
</pre>
</div>
<div class="section" id="id20">
<h1>20</h1>
<p>求 200! 的每位数字之和</p>
<pre class="code julia literal-block">
<span class="n">println</span><span class="p">(</span><span class="n">sum</span><span class="p">(</span><span class="n">map</span><span class="p">(</span><span class="n">c</span><span class="o">-></span><span class="n">c</span><span class="o">-</span><span class="sc">'0'</span><span class="p">,</span> <span class="n">collect</span><span class="p">(</span><span class="n">string</span><span class="p">(</span><span class="n">factorial</span><span class="p">(</span><span class="kt">BigInt</span><span class="p">(</span><span class="mi">100</span><span class="p">)))))))</span>
</pre>
</div>
<div class="section" id="id21">
<h1>21</h1>
<p>求 1000 以下满足 <tt class="docutils literal">a</tt> 的所有因数加起来等于 <tt class="docutils literal">b</tt> 且 <tt class="docutils literal">b</tt> 的所有因数加起来等于 <tt class="docutils literal">a</tt> 的 <tt class="docutils literal">a</tt>、<tt class="docutils literal">b</tt> 之和</p>
<pre class="code julia literal-block">
<span class="k">function</span> <span class="n">factors</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="k">for</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="kp">in</span> <span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">reduce</span><span class="p">(</span> <span class="n">vcat</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span>
<span class="p">[</span><span class="n">f</span> <span class="o">*</span> <span class="n">p</span><span class="o">^</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="n">n</span><span class="p">]</span> <span class="p">)</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">f</span>
<span class="k">end</span>
<span class="n">d</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">=</span> <span class="n">sum</span><span class="p">(</span><span class="n">factors</span><span class="p">(</span><span class="n">n</span><span class="p">))</span> <span class="o">-</span> <span class="n">n</span>
<span class="n">s</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">9999</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">d</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="k">if</span> <span class="mi">0</span><span class="o"><</span><span class="n">v</span><span class="o"><=</span><span class="mi">9999</span> <span class="o">&&</span> <span class="n">i!</span><span class="o">=</span><span class="n">v</span> <span class="o">&&</span> <span class="n">i</span><span class="o">==</span><span class="n">d</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">i</span> <span class="c"># v will be calculated later</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id22">
<h1>22</h1>
<p>求一个名字列表里面(每个名字的(字母序之和)与(它在列表中的字典序)的积)的和</p>
<pre class="code julia literal-block">
<span class="n">names</span><span class="o">=</span><span class="p">[</span> <span class="s">"MARY"</span><span class="p">,</span> <span class="o">...</span> <span class="p">,</span><span class="s">"ALONSO"</span><span class="p">]</span>
<span class="n">sort!</span><span class="p">(</span><span class="n">names</span><span class="p">)</span>
<span class="n">av</span><span class="p">(</span><span class="n">str</span><span class="p">)</span> <span class="o">=</span> <span class="n">sum</span><span class="p">(</span><span class="n">map</span><span class="p">(</span><span class="n">c</span><span class="o">-></span><span class="n">c</span><span class="o">-</span><span class="sc">'A'</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">collect</span><span class="p">(</span><span class="n">str</span><span class="p">)))</span>
<span class="n">assert</span><span class="p">(</span><span class="n">names</span><span class="p">[</span><span class="mi">938</span><span class="p">]</span><span class="o">==</span><span class="s">"COLIN"</span><span class="p">)</span>
<span class="n">s</span> <span class="o">=</span> <span class="kt">BigInt</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="n">length</span><span class="p">(</span><span class="n">names</span><span class="p">)</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">i</span> <span class="o">*</span> <span class="n">av</span><span class="p">(</span><span class="n">names</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id23">
<h1>23</h1>
<p>求所有(不能用两个(因子之和大于本身的数)之和表示的数字)的和</p>
<pre class="code julia literal-block">
<span class="k">function</span> <span class="n">factors</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="k">for</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="kp">in</span> <span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">reduce</span><span class="p">(</span> <span class="n">vcat</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span>
<span class="p">[</span><span class="n">f</span> <span class="o">*</span> <span class="n">p</span><span class="o">^</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="n">n</span><span class="p">]</span> <span class="p">)</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">f</span>
<span class="k">end</span>
<span class="n">d</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">=</span> <span class="n">sum</span><span class="p">(</span><span class="n">factors</span><span class="p">(</span><span class="n">n</span><span class="p">))</span> <span class="o">-</span> <span class="n">n</span>
<span class="n">isabundant</span> <span class="o">=</span> <span class="p">[</span><span class="kc">false</span> <span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">28123</span><span class="p">]</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">28123</span>
<span class="k">if</span> <span class="n">d</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">></span><span class="n">i</span>
<span class="n">isabundant</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">true</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">s</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">for</span> <span class="n">i</span><span class="o">=</span><span class="mi">2</span><span class="o">:</span><span class="mi">28123</span>
<span class="n">b</span> <span class="o">=</span> <span class="kc">false</span>
<span class="k">for</span> <span class="n">j</span><span class="o">=</span><span class="mi">1</span><span class="o">:</span><span class="n">int</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="n">i</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
<span class="k">if</span> <span class="n">isabundant</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&&</span> <span class="n">isabundant</span><span class="p">[</span><span class="n">i</span><span class="o">-</span><span class="n">j</span><span class="p">]</span>
<span class="n">b</span> <span class="o">=</span> <span class="kc">true</span>
<span class="k">break</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">if</span> <span class="o">!</span><span class="n">b</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">i</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id24">
<h1>24</h1>
<p>求 0 .. 9 的排列组合中字典序的第 1000000 位是什么</p>
<pre class="code julia literal-block">
<span class="n">ps</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="kp">in</span> <span class="n">permutations</span><span class="p">([</span><span class="mi">0</span><span class="o">:</span><span class="mi">9</span><span class="p">])]</span>
<span class="n">println</span><span class="p">(</span><span class="n">ps</span><span class="p">[</span><span class="mi">1000000</span><span class="p">])</span>
</pre>
</div>
<div class="section" id="id25">
<h1>25</h1>
<p>求斐波拉切数列从哪一位开始大于 1000 位数字</p>
<pre class="code julia literal-block">
<span class="n">a</span> <span class="o">=</span> <span class="kt">BigInt</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="kt">BigInt</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">c</span> <span class="o">=</span> <span class="kt">BigInt</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">3</span>
<span class="n">T</span> <span class="o">=</span> <span class="kt">BigInt</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">^</span><span class="mi">999</span>
<span class="k">while</span> <span class="kc">true</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">a</span><span class="o">+</span><span class="n">b</span>
<span class="k">if</span> <span class="n">c</span><span class="o">>=</span><span class="n">T</span>
<span class="k">break</span>
<span class="k">end</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">b</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">c</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id26">
<h1>26</h1>
<p>求在 [2..1000) 区间内使 1/d 的循环节最长的 d</p>
<pre class="code julia literal-block">
<span class="k">function</span> <span class="n">recurringCycleLen</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="c"># length of recurrying cycle of 1/n</span>
<span class="n">R</span> <span class="o">=</span> <span class="kt">Int</span><span class="p">[]</span> <span class="c"># all occured remainders</span>
<span class="n">r</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">c</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="n">len</span> <span class="o">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="k">while</span> <span class="n">r</span><span class="o">></span><span class="mi">0</span> <span class="o">&&</span> <span class="o">!</span><span class="kp">in</span><span class="p">(</span><span class="n">r</span><span class="p">,</span><span class="n">R</span><span class="p">)</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">int</span><span class="p">(</span><span class="n">floor</span><span class="p">(</span><span class="n">r</span><span class="o">/</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">r</span>
<span class="n">push!</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
<span class="n">r</span> <span class="o">-=</span> <span class="n">c</span> <span class="o">*</span> <span class="n">n</span>
<span class="k">end</span>
<span class="n">r</span> <span class="o">*=</span> <span class="mi">10</span>
<span class="k">end</span>
<span class="k">if</span> <span class="n">r</span><span class="o">==</span><span class="mi">0</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">length</span><span class="p">(</span><span class="n">R</span><span class="p">)</span><span class="o">-</span><span class="n">findfirst</span><span class="p">(</span><span class="n">R</span><span class="p">,</span><span class="n">r</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span>
<span class="k">end</span>
<span class="n">maxn</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">maxl</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">1000</span>
<span class="n">l</span> <span class="o">=</span> <span class="n">recurringCycleLen</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="k">if</span> <span class="n">l</span><span class="o">></span><span class="n">maxl</span>
<span class="n">maxn</span> <span class="o">=</span> <span class="n">i</span>
<span class="n">maxl</span> <span class="o">=</span> <span class="n">l</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">maxn</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id27">
<h1>27</h1>
<p>求使得 <span class="formula">∀<i>i</i> ∈ [0, <i>N</i>] <i>i</i><sup>2</sup> + <i>a</i>*<i>i</i> + <i>b</i><i>为质数</i></span> 中的 N 取值最大的 <span class="formula"><i>a</i>, <i>b</i></span> 的乘积，
其中 <span class="formula">|<i>a</i>| < 1000</span> 且 <span class="formula">|<i>b</i>| < 1000</span></p>
<pre class="code julia literal-block">
<span class="n">maxlen</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">maxa</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">maxb</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">a</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1000</span><span class="o">:</span><span class="mi">1000</span>
<span class="k">for</span> <span class="n">b</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1000</span><span class="o">:</span><span class="mi">1000</span>
<span class="n">len</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="n">isprime</span><span class="p">(</span><span class="n">n</span><span class="o">^</span><span class="mi">2</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
<span class="n">len</span><span class="o">+=</span><span class="mi">1</span>
<span class="n">n</span><span class="o">+=</span><span class="mi">1</span>
<span class="k">end</span>
<span class="k">if</span> <span class="n">len</span><span class="o">></span><span class="n">maxlen</span>
<span class="n">maxlen</span> <span class="o">=</span> <span class="n">len</span>
<span class="n">maxa</span> <span class="o">=</span> <span class="n">a</span>
<span class="n">maxb</span> <span class="o">=</span> <span class="n">b</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="s">"a = </span><span class="si">$maxa</span><span class="s">, b = </span><span class="si">$maxb</span><span class="s">, maxlen = </span><span class="si">$maxlen</span><span class="s">, a*b=</span><span class="si">$</span><span class="p">(</span><span class="n">maxa</span><span class="o">*</span><span class="n">maxb</span><span class="p">)</span><span class="s">"</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id28">
<h1>28</h1>
<p>求数字螺旋对角线上的值的和</p>
<pre class="code julia literal-block">
<span class="n">c</span> <span class="o">=</span> <span class="mi">1</span> <span class="c"># current</span>
<span class="n">s</span> <span class="o">=</span> <span class="mi">1</span> <span class="c"># sum</span>
<span class="n">step</span> <span class="o">=</span> <span class="mi">2</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">500</span>
<span class="k">for</span> <span class="n">j</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">4</span>
<span class="n">c</span> <span class="o">+=</span> <span class="n">step</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">c</span>
<span class="k">end</span>
<span class="n">step</span> <span class="o">+=</span> <span class="mi">2</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id29">
<h1>29</h1>
<p>求对于 <span class="formula">2 ≤ <i>a</i> ≤ 100</span> 且 <span class="formula">2 ≤ <i>b</i> ≤ 100</span>,
<span class="formula"><i>a</i><sup><i>b</i></sup></span> 一共有多少个不重复的值</p>
<p>任何一个理智的人都应该算出 <span class="formula">99 × 99</span> 然后减去重复的 …… 然而其实直接算也挺快的，不管了 ……</p>
<pre class="code julia literal-block">
<span class="n">li</span> <span class="o">=</span> <span class="p">[</span><span class="s">"4"</span><span class="o">=></span><span class="mi">1</span><span class="p">]</span>
<span class="k">for</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">2</span><span class="o">:</span><span class="mi">100</span>
<span class="k">for</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">2</span><span class="o">:</span><span class="mi">100</span>
<span class="n">li</span><span class="p">[</span><span class="n">string</span><span class="p">(</span><span class="kt">BigInt</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">^</span><span class="n">j</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">length</span><span class="p">(</span><span class="n">li</span><span class="p">))</span>
</pre>
</div>
<div class="section" id="id30">
<h1>30</h1>
<p>求所有可以用自身每位数字的五次方相加表示的整数</p>
<p>需要思考的问题只有 —— 上限是多少？</p>
<p>若我们把一个大于 0 的 n 位数字表示为 <span class="formula"><span class="limits"><span class="limit">∑</span></span><span class="scripts"><sup class="script"><i>n</i></sup><sub class="script"><i>i</i> = 1</sub></span><i>d</i><sub><i>i</i></sub> × 10<sup><i>n</i> − <i>i</i></sup></span>,
其中 <span class="formula"><i>d</i><sub><i>i</i></sub></span> 表示其第 i 位的数字，那么不难看出，
每位数字的五次方相加的值域是 <span class="formula">1 ≤ <span class="limits"><span class="limit">∑</span></span><span class="scripts"><sup class="script"><i>n</i></sup><sub class="script"><i>i</i> = 1</sub></span><i>d</i><sub><i>i</i></sub><sup>5</sup> ≤ 9<sup>5</sup> × <i>n</i></span>,
而其自身的值域则是 <span class="formula">10<sup><i>n</i> − 1</sup> ≤ <span class="limits"><span class="limit">∑</span></span><span class="scripts"><sup class="script"><i>n</i></sup><sub class="script"><i>i</i> = 1</sub></span><i>d</i><sub><i>i</i></sub> × 10<sup><i>n</i> − <i>i</i></sup> ≤ 10<sup><i>n</i></sup> − 1</span></p>
<p>于是，当 n>6 的时候， <span class="formula">9<sup>5</sup> × <i>n</i> < 10<sup><i>n</i> − 1</sup></span>, 从这里开始往后不可能有解了</p>
<pre class="code julia literal-block">
<span class="c"># when n>6, 9^5*n < 10^(n-1)</span>
<span class="c"># so set the upper limit to 9^5*6 is safe</span>
<span class="n">ps</span> <span class="o">=</span> <span class="n">map</span><span class="p">(</span><span class="n">x</span><span class="o">-></span><span class="n">x</span><span class="o">^</span><span class="mi">5</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="o">:</span><span class="mi">9</span><span class="p">])</span>
<span class="n">s</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">2</span><span class="o">:</span><span class="mi">9</span><span class="o">^</span><span class="mi">5</span><span class="o">*</span><span class="mi">6</span>
<span class="k">if</span> <span class="n">sum</span><span class="p">(</span><span class="n">map</span><span class="p">(</span><span class="n">c</span><span class="o">-></span><span class="n">ps</span><span class="p">[</span><span class="n">c</span><span class="o">-</span><span class="sc">'0'</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="n">collect</span><span class="p">(</span><span class="n">string</span><span class="p">(</span><span class="n">i</span><span class="p">))))</span> <span class="o">==</span> <span class="n">i</span>
<span class="c"># println(i)</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">i</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id31">
<h1>31</h1>
<p>求使用面值 1p, 2p, 5p, 10p, 20p, 50p, ￡1, ￡2 面值的硬币，一共有多少种方式组成 ￡2,
其中 ￡1 = 100p</p>
<p>经典的 DP 问题 ——若是已知不使用面值 x 的硬币，数字 a 有 N[a] 种表示法，
显然加上 x 面值硬币之后，若 a>=x，数字 a 的表示方法将是 N[a] + N[a-x]，否则依然是 N[a]。
于是可以这么递推出来 ——</p>
<pre class="code julia literal-block">
<span class="n">cs</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">5</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">20</span><span class="p">,</span><span class="mi">50</span><span class="p">,</span><span class="mi">100</span><span class="p">,</span><span class="mi">200</span><span class="p">]</span>
<span class="n">ws</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">200</span><span class="p">]</span>
<span class="k">for</span> <span class="n">c</span> <span class="kp">in</span> <span class="n">cs</span>
<span class="n">ws</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span> <span class="c"># just c</span>
<span class="k">for</span> <span class="n">v</span> <span class="kp">in</span> <span class="n">c</span><span class="o">+</span><span class="mi">1</span><span class="o">:</span><span class="mi">200</span>
<span class="n">ws</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">+=</span> <span class="n">ws</span><span class="p">[</span><span class="n">v</span><span class="o">-</span><span class="n">c</span><span class="p">]</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">ws</span><span class="p">[</span><span class="mi">200</span><span class="p">])</span>
</pre>
</div>
<div class="section" id="id32">
<h1>32</h1>
<p>找出所有 <span class="formula"><i>a</i> × <i>b</i> = <i>c</i></span> 且 <span class="formula"><i>a</i>, <i>b</i>, <i>c</i></span> 每一位数字组成的列表正好包含 1 .. 9 每个数字各一次</p>
<pre class="code julia literal-block">
<span class="n">ps</span> <span class="o">=</span> <span class="kt">Int</span><span class="p">[]</span>
<span class="n">one2nine</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="s">"123456789"</span><span class="p">)</span>
<span class="k">function</span> <span class="n">hasrep</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">s</span><span class="o">=</span><span class="n">string</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">2</span><span class="o">:</span><span class="n">length</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="k">if</span> <span class="n">s</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">s</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="k">return</span> <span class="kc">true</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">return</span> <span class="kc">false</span>
<span class="k">end</span>
<span class="k">for</span> <span class="n">i</span><span class="o">=</span><span class="mi">2</span><span class="o">:</span><span class="mi">2000</span>
<span class="k">if</span> <span class="n">hasrep</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="k">continue</span>
<span class="k">end</span>
<span class="k">for</span> <span class="n">j</span><span class="o">=</span><span class="n">i</span><span class="o">:</span><span class="mi">2000</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">i</span><span class="o">*</span><span class="n">j</span>
<span class="c">#if hasrep(j)</span>
<span class="c"># continue</span>
<span class="c">#end</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">string</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
<span class="k">if</span> <span class="n">length</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">==</span><span class="mi">9</span> <span class="o">&&</span> <span class="n">sort</span><span class="p">(</span><span class="n">collect</span><span class="p">(</span><span class="n">s</span><span class="p">))</span><span class="o">==</span><span class="n">one2nine</span>
<span class="n">push!</span><span class="p">(</span><span class="n">ps</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
<span class="n">println</span><span class="p">(</span><span class="s">"</span><span class="si">$i</span><span class="s"> x </span><span class="si">$j</span><span class="s"> = </span><span class="si">$p</span><span class="s">"</span><span class="p">);</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">sort!</span><span class="p">(</span><span class="n">ps</span><span class="p">)</span>
<span class="n">s</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span><span class="o">=</span><span class="mi">2</span><span class="o">:</span><span class="n">length</span><span class="p">(</span><span class="n">ps</span><span class="p">)</span>
<span class="k">if</span> <span class="n">ps</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">ps</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">ps</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="k">end</span>
<span class="k">end</span>
<span class="k">if</span> <span class="n">ps</span><span class="p">[</span><span class="k">end</span><span class="p">]</span><span class="o">!=</span><span class="n">ps</span><span class="p">[</span><span class="k">end</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">ps</span><span class="p">[</span><span class="k">end</span><span class="p">]</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id33">
<h1>33</h1>
<p>找出分子分母各是两位数且能够从分子分母上各删除同一个数字其值不变的分数</p>
<p><span class="formula">11 ⁄ 22 = 1 ⁄ 2</span> 或是 <span class="formula">10 ⁄ 20 = 1 ⁄ 2</span> 这种显然成立的除外</p>
<pre class="code julia literal-block">
<span class="n">fs</span> <span class="o">=</span> <span class="kt">Rational</span><span class="p">{</span><span class="kt">Int64</span><span class="p">}[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">12</span><span class="o">:</span><span class="mi">98</span>
<span class="k">for</span> <span class="n">j</span> <span class="kp">in</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="o">:</span><span class="mi">99</span>
<span class="c"># i = a*10 + b</span>
<span class="c"># j = c*10 + d</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">int</span><span class="p">(</span><span class="n">floor</span><span class="p">(</span><span class="n">i</span><span class="o">/</span><span class="mi">10</span><span class="p">))</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">i</span><span class="o">%</span><span class="mi">10</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">int</span><span class="p">(</span><span class="n">floor</span><span class="p">(</span><span class="n">j</span><span class="o">/</span><span class="mi">10</span><span class="p">))</span>
<span class="n">d</span> <span class="o">=</span> <span class="n">j</span><span class="o">%</span><span class="mi">10</span>
<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">d</span>
<span class="k">continue</span>
<span class="k">end</span>
<span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="o">//</span><span class="n">j</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">a</span><span class="o">==</span><span class="n">d</span> <span class="o">||</span>
<span class="n">i</span><span class="o">//</span><span class="n">j</span> <span class="o">==</span> <span class="n">a</span><span class="o">//</span><span class="n">d</span> <span class="o">&&</span> <span class="n">b</span><span class="o">==</span><span class="n">c</span><span class="p">)</span>
<span class="n">push!</span><span class="p">(</span><span class="n">fs</span><span class="p">,</span> <span class="n">i</span><span class="o">//</span><span class="n">j</span><span class="p">)</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">prod</span><span class="p">(</span><span class="n">fs</span><span class="p">))</span>
</pre>
</div>
<div class="section" id="id34">
<h1>34</h1>
<p>找出所有(每一位数字的阶乘的和等于自身)的数字的和</p>
<p>与 <a class="reference internal" href="#id30">30</a> 题相似的是，只要找到值域就好算了</p>
<pre class="code julia literal-block">
<span class="c"># factorial(9) = 362880</span>
<span class="c"># for number has n digits, 1 <= sum of its factorial digits <= 362880n</span>
<span class="c"># for number has n digits, 10^(n-1) <= its value < 10^n</span>
<span class="n">fac</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span><span class="o">=></span><span class="n">factorial</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="kp">in</span> <span class="mi">0</span><span class="o">:</span><span class="mi">9</span><span class="p">]</span>
<span class="k">function</span> <span class="n">digifacsum</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">s</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="n">n</span><span class="o">>=</span><span class="mi">1</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">fac</span><span class="p">[</span><span class="n">n</span><span class="o">%</span><span class="mi">10</span><span class="p">]</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">int</span><span class="p">(</span><span class="n">floor</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="mi">10</span><span class="p">))</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">s</span>
<span class="k">end</span>
<span class="n">s</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">3</span><span class="o">:</span><span class="n">fac</span><span class="p">[</span><span class="mi">9</span><span class="p">]</span><span class="o">*</span><span class="mi">8</span>
<span class="k">if</span> <span class="n">digifacsum</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">==</span><span class="n">i</span>
<span class="n">println</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">i</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id35">
<h1>35</h1>
<p><span class="formula">10<sup>6</sup></span> 以下有几个质数随意循环移位还是质数</p>
<pre class="code julia literal-block">
<span class="k">function</span> <span class="n">tonum</span><span class="p">(</span><span class="n">carr</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">d</span> <span class="kp">in</span> <span class="n">carr</span>
<span class="n">v</span> <span class="o">*=</span> <span class="mi">10</span>
<span class="n">v</span> <span class="o">+=</span> <span class="n">d</span><span class="o">-</span><span class="sc">'0'</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">v</span>
<span class="k">end</span>
<span class="k">function</span> <span class="n">circular</span><span class="p">(</span><span class="n">prm</span><span class="p">)</span>
<span class="n">res</span> <span class="o">=</span> <span class="kt">Int</span><span class="p">[]</span>
<span class="n">col</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="n">string</span><span class="p">(</span><span class="n">prm</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">2</span><span class="o">:</span><span class="n">length</span><span class="p">(</span><span class="n">col</span><span class="p">)</span>
<span class="n">push!</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">tonum</span><span class="p">(</span><span class="n">vcat</span><span class="p">(</span><span class="n">col</span><span class="p">[</span><span class="n">i</span><span class="o">:</span><span class="k">end</span><span class="p">],</span> <span class="n">col</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])))</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">res</span>
<span class="k">end</span>
<span class="n">cnt</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">p</span> <span class="kp">in</span> <span class="n">primes</span><span class="p">(</span><span class="mi">1000000</span><span class="p">)</span>
<span class="n">iscircular</span> <span class="o">=</span> <span class="kc">true</span>
<span class="k">for</span> <span class="n">x</span> <span class="kp">in</span> <span class="n">circular</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="k">if</span> <span class="o">!</span><span class="n">isprime</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">iscircular</span> <span class="o">=</span> <span class="kc">false</span>
<span class="k">break</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">if</span> <span class="n">iscircular</span>
<span class="c"># println(p)</span>
<span class="n">cnt</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">cnt</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id36">
<h1>36</h1>
<p>求 <span class="formula">10<sup>6</sup></span> 以下在十进制与二进制下都是回文数的数字之和</p>
<pre class="code julia literal-block">
<span class="k">function</span> <span class="n">ispalindrome</span><span class="p">(</span><span class="n">s</span><span class="o">::</span><span class="n">String</span><span class="p">)</span>
<span class="k">if</span> <span class="n">length</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o"><=</span><span class="mi">2</span>
<span class="k">return</span> <span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="n">s</span><span class="p">[</span><span class="k">end</span><span class="p">]</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">==</span><span class="n">s</span><span class="p">[</span><span class="k">end</span><span class="p">]</span> <span class="o">&&</span> <span class="n">ispalindrome</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">2</span><span class="o">:</span><span class="k">end</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">sum</span><span class="p">(</span><span class="n">filter</span><span class="p">(</span><span class="n">i</span><span class="o">-></span><span class="n">ispalindrome</span><span class="p">(</span><span class="n">string</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="o">&&</span> <span class="n">ispalindrome</span><span class="p">(</span><span class="n">base</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">i</span><span class="p">)),</span>
<span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">1000000</span><span class="p">])))</span>
</pre>
</div>
<div class="section" id="id37">
<h1>37</h1>
<p>求从左到由每次删掉一位数字依然是质数的质数之和，已知一共有 11 个这样的质数</p>
<pre class="code julia literal-block">
<span class="n">ps</span> <span class="o">=</span> <span class="n">primes</span><span class="p">(</span><span class="mi">1000000</span><span class="p">)</span> <span class="c"># just guess</span>
<span class="k">function</span> <span class="n">istruncatable</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">10</span>
<span class="k">return</span> <span class="kc">false</span>
<span class="k">end</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">n</span>
<span class="k">while</span> <span class="n">x</span><span class="o">>=</span><span class="mi">1</span>
<span class="k">if</span> <span class="o">!</span><span class="n">isprime</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">return</span> <span class="kc">false</span> <span class="k">end</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">int</span><span class="p">(</span><span class="n">floor</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">10</span><span class="p">))</span>
<span class="k">end</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">string</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="o">=</span><span class="mi">2</span><span class="o">:</span><span class="n">length</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="k">if</span> <span class="o">!</span><span class="n">isprime</span><span class="p">(</span><span class="n">int</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="o">:</span><span class="k">end</span><span class="p">]))</span> <span class="k">return</span> <span class="kc">false</span> <span class="k">end</span>
<span class="k">end</span>
<span class="k">return</span> <span class="kc">true</span>
<span class="k">end</span>
<span class="n">cnt</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">s</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">p</span> <span class="kp">in</span> <span class="n">ps</span>
<span class="k">if</span> <span class="n">istruncatable</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">println</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">p</span>
<span class="o">++</span><span class="n">cnt</span>
<span class="k">if</span> <span class="n">cnt</span><span class="o">==</span><span class="mi">11</span>
<span class="k">break</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id38">
<h1>38</h1>
<pre class="code julia literal-block">
<span class="n">ispandigital</span><span class="p">(</span><span class="n">s</span><span class="o">::</span><span class="n">String</span><span class="p">)</span> <span class="o">=</span> <span class="n">sort</span><span class="p">(</span><span class="n">collect</span><span class="p">(</span><span class="n">s</span><span class="p">))</span> <span class="o">==</span> <span class="n">collect</span><span class="p">(</span><span class="s">"123456789"</span><span class="p">)</span>
<span class="o">+</span><span class="p">(</span><span class="n">s1</span><span class="o">::</span><span class="n">String</span><span class="p">,</span><span class="n">s2</span><span class="o">::</span><span class="n">String</span><span class="p">)</span> <span class="o">=</span> <span class="n">string</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span><span class="n">s2</span><span class="p">)</span>
<span class="k">function</span> <span class="n">guesspandigital</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">string</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">i</span> <span class="o">=</span> <span class="mi">2</span>
<span class="k">while</span> <span class="n">length</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o"><</span><span class="mi">9</span>
<span class="n">s</span> <span class="o">+=</span> <span class="n">string</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">i</span><span class="p">)</span>
<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">s</span>
<span class="k">end</span>
<span class="n">ispandigitalmultiple</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">=</span> <span class="n">ispandigital</span><span class="p">(</span><span class="n">guesspandigital</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="n">largest</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">p</span> <span class="kp">in</span> <span class="mi">2</span><span class="o">:</span><span class="mi">10000</span> <span class="c"># 10000 -> 1000020000</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">guesspandigital</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="k">if</span> <span class="n">ispandigital</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">&&</span> <span class="n">largest</span><span class="o"><</span><span class="n">int</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="n">largest</span> <span class="o">=</span> <span class="n">int</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="k">end</span>
<span class="n">p</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">largest</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id39">
<h1>39</h1>
<p>问对于 <span class="formula"><i>p</i> ≤ 1000</span>, <span class="formula"><i>p</i></span> 取值多少的时候方程</p>
<div class="formula">
<span class="array"><span class="arrayrow"><span class="bracket align-l">⎧</span></span><span class="arrayrow"><span class="bracket align-l">⎪</span></span><span class="arrayrow"><span class="bracket align-l">⎨</span></span><span class="arrayrow"><span class="bracket align-l">⎪</span></span><span class="arrayrow"><span class="bracket align-l">⎩</span></span></span><span class="bracketcases">
<span class="arrayrow">
<span class="case align-l">
<i>a</i><sup>2</sup> + <i>b</i><sup>2</sup> = <i>c</i><sup>2</sup>
</span>
</span>
<span class="arrayrow">
<span class="case align-l">
</span>
</span>
<span class="arrayrow">
<span class="case align-l">
<i>a</i> < <i>b</i> < <i>c</i>
</span>
</span>
<span class="arrayrow">
<span class="case align-l">
</span>
</span>
<span class="arrayrow">
<span class="case align-l">
<i>a</i> + <i>b</i> + <i>c</i> = <i>p</i>
</span>
</span>
</span>
</div>
<p>有最多的解</p>
<pre class="code julia literal-block">
<span class="k">function</span> <span class="n">numsolution</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
<span class="n">cnt</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">a</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">300</span>
<span class="k">for</span> <span class="n">b</span> <span class="kp">in</span> <span class="n">a</span><span class="o">+</span><span class="mi">1</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="p">)</span><span class="o">/</span><span class="mi">2</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</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="p">)</span>
<span class="k">if</span> <span class="n">isinteger</span><span class="p">(</span><span class="n">c</span><span class="p">)</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">c</span> <span class="o">==</span> <span class="n">p</span>
<span class="n">cnt</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">return</span> <span class="n">cnt</span>
<span class="k">end</span>
<span class="n">x</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="kp">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">1000</span>
<span class="n">ns</span> <span class="o">=</span> <span class="n">numsolution</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="k">if</span> <span class="n">ns</span><span class="o">></span><span class="n">n</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">ns</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">i</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="n">println</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
</pre>
</div>
<div class="section" id="id40">
<h1>40</h1>
<p>由于数据量太小（对于我的计算机），我就直接算出来了</p>
<p>出题者意图肯定不是希望这么作弊的 - -</p>
<pre class="code julia literal-block">
<span class="o">+</span><span class="p">(</span><span class="n">s1</span><span class="o">::</span><span class="n">String</span><span class="p">,</span><span class="n">s2</span><span class="o">::</span><span class="n">String</span><span class="p">)</span> <span class="o">=</span> <span class="n">string</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span><span class="n">s2</span><span class="p">)</span>
<span class="n">s</span> <span class="o">=</span> <span class="n">sum</span><span class="p">(</span><span class="n">map</span><span class="p">(</span><span class="n">string</span><span class="p">,[</span><span class="mi">1</span><span class="o">:</span><span class="mi">1000000</span><span class="p">]))</span>
<span class="n">println</span><span class="p">(</span><span class="n">prod</span><span class="p">(</span><span class="n">map</span><span class="p">(</span><span class="n">i</span><span class="o">-></span><span class="n">s</span><span class="p">[</span><span class="mi">10</span><span class="o">^</span><span class="n">i</span><span class="p">]</span><span class="o">-</span><span class="sc">'0'</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="o">:</span><span class="mi">6</span><span class="p">])))</span>
</pre>
</div>
<div class="section" id="id41">
<h1>41</h1>
<p>求最大的包含 <span class="formula">[1, <i>n</i>]</span> 每个数字的 n 位质数</p>
<p>显然 n 不会超过 9，直接从后往前找包含 <span class="formula">[1, <i>n</i>]</span> 每个数字的数判断一下</p>
<pre class="code julia literal-block">
<span class="n">ps</span> <span class="o">=</span> <span class="kt">Int</span><span class="p">[]</span>
<span class="n">tonum</span><span class="p">(</span><span class="n">li</span><span class="p">)</span> <span class="o">=</span> <span class="n">sum</span><span class="p">(</span><span class="n">map</span><span class="p">(</span><span class="n">i</span><span class="o">-></span><span class="n">li</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="mi">10</span><span class="o">^</span><span class="p">(</span><span class="n">length</span><span class="p">(</span><span class="n">li</span><span class="p">)</span><span class="o">-</span><span class="n">i</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="n">length</span><span class="p">(</span><span class="n">li</span><span class="p">)]))</span>
<span class="k">for</span> <span class="n">d</span> <span class="kp">in</span> <span class="mi">9</span><span class="o">:-</span><span class="mi">1</span><span class="o">:</span><span class="mi">2</span>
<span class="n">found</span> <span class="o">=</span> <span class="kc">false</span>
<span class="k">for</span> <span class="n">p</span> <span class="kp">in</span> <span class="n">permutations</span><span class="p">([</span><span class="n">d</span><span class="o">:-</span><span class="mi">1</span><span class="o">:</span><span class="mi">1</span><span class="p">])</span>
<span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">tonum</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
<span class="n">push!</span><span class="p">(</span><span class="n">ps</span><span class="p">,</span><span class="n">tonum</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
<span class="n">found</span> <span class="o">=</span> <span class="kc">true</span>
<span class="k">break</span>
<span class="k">end</span>
<span class="k">end</span>
<span class="k">if</span> <span class="n">found</span>
<span class="k">break</span>