# Problem 289: Eulerian Cycles
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 1010.