## Eulerian Cycles

### Problem 289

Published on Friday, 23rd April 2010, 01:00 pm; Solved by 325; Difficulty rating: 90%Let C(`x`,`y`) be a circle passing through the points (`x`, `y`), (`x`, `y`+1), (`x`+1, `y`) and (`x`+1, `y`+1).

For positive integers m and n, let E(`m`,`n`) be a configuration which consists of the `m`·`n` circles:

{ C(`x`,`y`): 0 ≤ `x` < `m`, 0 ≤ `y` < `n`, `x` and `y` are integers }

An Eulerian cycle on E(`m`,`n`) is a closed path that passes through each arc exactly once.

Many such paths are possible on E(`m`,`n`), but we are only interested in those which are not self-crossing:
A non-crossing path just touches itself at lattice points, but it never crosses itself.

The image below shows E(3,3) and an example of an Eulerian non-crossing path.

Let L(`m`,`n`) be the number of Eulerian non-crossing paths on E(`m`,`n`).

For example, L(1,2) = 2, L(2,2) = 37 and L(3,3) = 104290.

Find L(6,10) mod 10^{10}.