PAT/source/html/a1128.html

N Queens Puzzle
<div class="ques-view"><p>The "eight queens puzzle" is the problem of placing eight chess queens on an <span class="katex"><span class="katex-mathml"><math><mrow><mn>8</mn><mo>×</mo><mn>8</mn></mrow>8\times 8</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">8</span><span class="mbin">×</span><span class="mord mathrm">8</span></span></span></span> chessboard so that no two queens threaten each other. Thus, a solution requires that no two queens share the same row, column, or diagonal. The eight queens puzzle is an example of the more general <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> queens problem of placing <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> non-attacking queens on an <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi><mo>×</mo><mi>N</mi></mrow>N\times N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span><span class="mbin">×</span><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span>  chessboard. (From Wikipedia - "Eight queens puzzle".)</p>
<p>Here you are NOT asked to solve the puzzles. Instead, you are supposed to judge whether or not a given configuration of the chessboard is a solution. To simplify the representation of a chessboard, let us assume that no two queens will be placed in the same column. Then a configuration can be represented by a simple integer sequence <span class="katex"><span class="katex-mathml"><math><mrow><mo>(</mo><msub><mi>Q</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>Q</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mo separator="true">,</mo><msub><mi>Q</mi><mi>N</mi></msub><mo>)</mo></mrow>(Q_1, Q_2, \cdots , Q_N)</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord"><span class="mord mathit">Q</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="mpunct">,</span><span class="mord"><span class="mord mathit">Q</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">2</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="mpunct">,</span><span class="minner">⋯</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">Q</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:0.10903em;">N</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="mclose">)</span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow>Q_i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">Q</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></span></span> is the row number of the queen in the <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>-th column. For example, Figure 1 can be represented by (4, 6, 8, 2, 7, 1, 3, 5) and it is indeed a solution to the 8 queens puzzle; while Figure 2 can be represented by (4, 6, 7, 2, 8, 1, 9, 5, 3) and is NOT a 9 queens' solution.</p>
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<th style="text-align:center"><img alt="8q.jpg" src="https://images.ptausercontent.com/7d0443cf-5c19-4494-98a6-0f0f54894eaa.jpg"/></th>
<th></th>
<th style="text-align:center"><img alt="9q.jpg" src="https://images.ptausercontent.com/d187e37a-4eb8-4215-8e2c-040a73c5c8d8.jpg"/></th>
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<td style="text-align:center">Figure 1</td>
<td></td>
<td style="text-align:center">Figure 2</td>
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</tbody>
</table>
<h3 id="input-specification-">Input Specification:</h3>
<p>Each input file contains several test cases. The first line gives an integer <span class="katex"><span class="katex-mathml"><math><mrow><mi>K</mi></mrow>K</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:0.07153em;">K</span></span></span></span> (<span class="katex"><span class="katex-mathml"><math><mrow><mn>1</mn><mo>&lt;</mo><mi>K</mi><mo>≤</mo><mn>2</mn><mn>0</mn><mn>0</mn></mrow>1&lt;K\le 200</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span><span class="mrel">&lt;</span><span class="mord mathit" style="margin-right:0.07153em;">K</span><span class="mrel">≤</span><span class="mord mathrm">2</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span></span></span></span>). Then <span class="katex"><span class="katex-mathml"><math><mrow><mi>K</mi></mrow>K</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:0.07153em;">K</span></span></span></span> lines follow, each gives a configuration in the format "<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> <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>Q</mi><mn>1</mn></msub></mrow>Q_1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">Q</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></span></span> <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>Q</mi><mn>2</mn></msub></mrow>Q_2</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">Q</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">2</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> ... <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>Q</mi><mi>N</mi></msub></mrow>Q_N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">Q</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:0.10903em;">N</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>", where <span class="katex"><span class="katex-mathml"><math><mrow><mn>4</mn><mo>≤</mo><mi>N</mi><mo>≤</mo><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn></mrow>4\le N\le 1000</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">4</span><span class="mrel">≤</span><span class="mord mathit" style="margin-right:0.10903em;">N</span><span class="mrel">≤</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span><span class="mord mathrm">0</span></span></span></span> and it is guaranteed that <span class="katex"><span class="katex-mathml"><math><mrow><mn>1</mn><mo>≤</mo><msub><mi>Q</mi><mi>i</mi></msub><mo>≤</mo><mi>N</mi></mrow>1\le Q_i\le N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span><span class="mrel">≤</span><span class="mord"><span class="mord mathit">Q</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 mathit" style="margin-right:0.10903em;">N</span></span></span></span> for all <span class="katex"><span class="katex-mathml"><math><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mo>⋯</mo><mo separator="true">,</mo><mi>N</mi></mrow>i=1, \cdots , N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="minner">⋯</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span>. The numbers are separated by spaces.</p>
<h3 id="output-specification-">Output Specification:</h3>
<p>For each configuration, if it is a solution to the <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> queens problem, print <code>YES</code> in a line; or <code>NO</code> if not.</p>
<h3 id="sample-input-">Sample Input:</h3>
<pre><code class="lang-in">4
8 4 6 8 2 7 1 3 5
9 4 6 7 2 8 1 9 5 3
6 1 5 2 6 4 3
5 1 3 5 2 4
</code></pre>
<h3 id="sample-output-">Sample Output:</h3>
<pre><code class="lang-out">YES
NO
NO
YES
</code></pre>
</div>