PAT/source/html/b1079.html

延迟的回文数
<div class="ques-view"><p>给定一个 <span class="katex"><span class="katex-mathml"><math><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow>k+1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span> 位的正整数 <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span>，写成 <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>a</mi><mi>k</mi></msub><mo>⋯</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>a</mi><mn>0</mn></msub></mrow>a_k \cdots a_1 a_0</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="minner">⋯</span><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> 的形式，其中对所有 <span class="katex"><span class="katex-mathml"><math><mrow><mi>i</mi></mrow>i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.65952em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span></span></span></span> 有 <span class="katex"><span class="katex-mathml"><math><mrow><mn>0</mn><mo>≤</mo><msub><mi>a</mi><mi>i</mi></msub><mo>&lt;</mo><mn>1</mn><mn>0</mn></mrow>0 \le a_i &lt; 10</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">0</span><span class="mrel">≤</span><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mrel">&lt;</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>a</mi><mi>k</mi></msub><mo>&gt;</mo><mn>0</mn></mrow>a_k &gt; 0</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mrel">&gt;</span><span class="mord mathrm">0</span></span></span></span>。<span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span> 被称为一个<strong>回文数</strong>，当且仅当对所有 <span class="katex"><span class="katex-mathml"><math><mrow><mi>i</mi></mrow>i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.65952em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span></span></span></span> 有 <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>=</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mi>i</mi></mrow></msub></mrow>a_i = a_{k-i}</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mrel">=</span><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mathit mtight">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>。零也被定义为一个回文数。</p>
<p>非回文数也可以通过一系列操作变出回文数。首先将该数字逆转，再将逆转数与该数相加，如果和还不是一个回文数，就重复这个逆转再相加的操作，直到一个回文数出现。如果一个非回文数可以变出回文数，就称这个数为<strong>延迟的回文数</strong>。（定义翻译自 <a href="https://en.wikipedia.org/wiki/Palindromic_number">https://en.wikipedia.org/wiki/Palindromic_number</a> ）</p>
<p>给定任意一个正整数，本题要求你找到其变出的那个回文数。</p>
<h3 id="-">输入格式：</h3>
<p>输入在一行中给出一个不超过1000位的正整数。</p>
<h3 id="-">输出格式：</h3>
<p>对给定的整数，一行一行输出其变出回文数的过程。每行格式如下</p>
<pre><code>A + B = C
</code></pre><p>其中 <code>A</code> 是原始的数字，<code>B</code> 是 <code>A</code> 的逆转数，<code>C</code> 是它们的和。<code>A</code> 从输入的整数开始。重复操作直到 <code>C</code> 在 10 步以内变成回文数，这时在一行中输出 <code>C is a palindromic number.</code>；或者如果 10 步都没能得到回文数，最后就在一行中输出 <code>Not found in 10 iterations.</code>。</p>
<h3 id="-1-">输入样例 1：</h3>
<pre><code class="lang-in">97152
</code></pre>
<h3 id="-1-">输出样例 1：</h3>
<pre><code class="lang-out">97152 + 25179 = 122331
122331 + 133221 = 255552
255552 is a palindromic number.
</code></pre>
<h3 id="-2-">输入样例 2：</h3>
<pre><code class="lang-in">196
</code></pre>
<h3 id="-2-">输出样例 2：</h3>
<pre><code class="lang-out">196 + 691 = 887
887 + 788 = 1675
1675 + 5761 = 7436
7436 + 6347 = 13783
13783 + 38731 = 52514
52514 + 41525 = 94039
94039 + 93049 = 187088
187088 + 880781 = 1067869
1067869 + 9687601 = 10755470
10755470 + 07455701 = 18211171
Not found in 10 iterations.
</code></pre>
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