84 lines
5.6 KiB
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84 lines
5.6 KiB
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<h2>Eulerian Cycles</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 289</h3><span style="width:300px;color:#666;">Published on Friday, 23rd April 2010, 01:00 pm; Solved by 348</span></div>
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<p>Let C(<var>x</var>,<var>y</var>) be a circle passing through the points (<var>x</var>, <var>y</var>), (<var>x</var>, <var>y</var>+1), (<var>x</var>+1, <var>y</var>) and (<var>x</var>+1, <var>y</var>+1).</p>
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<p>For positive integers m and n, let E(<var>m</var>,<var>n</var>) be a configuration which consists of the <var>m</var>·<var>n</var> circles:<br />
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{ C(<var>x</var>,<var>y</var>): 0 ≤ <var>x</var> < <var>m</var>, 0 ≤ <var>y</var> < <var>n</var>, <var>x</var> and <var>y</var> are integers }</p>
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<p>An Eulerian cycle on E(<var>m</var>,<var>n</var>) is a closed path that passes through each arc exactly once.<br />
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Many such paths are possible on E(<var>m</var>,<var>n</var>), but we are only interested in those which are not self-crossing:
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A non-crossing path just touches itself at lattice points, but it never crosses itself.</p>
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<p>The image below shows E(3,3) and an example of an Eulerian non-crossing path.<br />
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<div align='center'><img src="project/images/p_289_euler.gif" /></div></p>
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<p>Let L(<var>m</var>,<var>n</var>) be the number of Eulerian non-crossing paths on E(<var>m</var>,<var>n</var>).<br />
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For example, L(1,2) = 2, L(2,2) = 37 and L(3,3) = 104290.</p>
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<p>Find L(6,10) mod 10<sup>10</sup>.</p>
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