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<!DOCTYPE html>
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<title>Problem 61 - Project Euler</title>
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<h2>Cyclical figurate numbers</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 61</h3><span style="width:300px;color:#666;">Published on Friday, 16th January 2004, 06:00 pm; Solved by 11199</span></div>
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<div class="problem_content" role="problem">
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<p>Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:</p>
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<table>
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<tr>
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<td>Triangle</td>
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<td> </td>
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<td>P<sub>3,<i>n</i></sub>=<i>n</i>(<i>n</i>+1)/2</td>
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<td> </td>
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<td>1, 3, 6, 10, 15, ...</td>
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</tr>
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<tr>
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<td>Square</td>
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<td> </td>
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<td>P<sub>4,<i>n</i></sub>=<i>n</i><sup>2</sup></td>
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<td> </td>
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<td>1, 4, 9, 16, 25, ...</td>
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</tr>
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<tr>
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<td>Pentagonal</td>
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<td> </td>
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<td>P<sub>5,<i>n</i></sub>=<i>n</i>(3<i>n</i><img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />1)/2</td>
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<td> </td>
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<td>1, 5, 12, 22, 35, ...</td>
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</tr>
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<tr>
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<td>Hexagonal</td>
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<td> </td>
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<td>P<sub>6,<i>n</i></sub>=<i>n</i>(2<i>n</i><img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />1)</td>
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<td> </td>
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<td>1, 6, 15, 28, 45, ...</td>
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</tr>
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<tr>
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<td>Heptagonal</td>
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<td> </td>
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<td>P<sub>7,<i>n</i></sub>=<i>n</i>(5<i>n</i><img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />3)/2</td>
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<td> </td>
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<td>1, 7, 18, 34, 55, ...</td>
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</tr>
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<tr>
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<td>Octagonal</td>
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<td> </td>
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<td>P<sub>8,<i>n</i></sub>=<i>n</i>(3<i>n</i><img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />2)</td>
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<td> </td>
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<td>1, 8, 21, 40, 65, ...</td>
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</tr>
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</table>
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<p>The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.</p>
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<ol>
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<li>The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).</li>
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<li>Each polygonal type: triangle (P<sub>3,127</sub>=8128), square (P<sub>4,91</sub>=8281), and pentagonal (P<sub>5,44</sub>=2882), is represented by a different number in the set.</li>
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<li>This is the only set of 4-digit numbers with this property.</li>
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</ol>
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<p>Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.</p>
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