92 lines
5.2 KiB
HTML
92 lines
5.2 KiB
HTML
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<h2>Cyclic paths on Sierpiński graphs
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</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 312</h3><span style="width:300px;color:#666;">Published on Sunday, 28th November 2010, 01:00 am; Solved by 476</span></div>
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<p>- A <b>Sierpiński graph</b> of order-1 (<var>S</var><sub>1</sub>) is an equilateral triangle.<br />
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- <var>S</var><sub><var>n</var>+1</sub> is obtained from <var>S</var><sub><var>n</var></sub> by positioning three copies of <var>S</var><sub><var>n</var></sub> so that every pair of copies has one common corner.
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<div align='center'><img src="project/images/p_312_sierpinskyAt.gif" /></div>
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<p>Let C(<var>n</var>) be the number of cycles that pass exactly once through all the vertices of <var>S</var><sub><var>n</var></sub>.<br />
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For example, C(3) = 8 because eight such cycles can be drawn on <var>S</var><sub>3</sub>, as shown below:
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</p>
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<div align='center'><img src="project/images/p_312_sierpinsky8t.gif" /></div>
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<p>It can also be verified that :<br />
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C(1) = C(2) = 1<br />
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C(5) = 71328803586048<br />
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C(10 000) mod 10<sup>8</sup> = 37652224<br />
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C(10 000) mod 13<sup>8</sup> = 617720485<br />
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</p>
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<p>Find C(C(C(10 000))) mod 13<sup>8</sup>.
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