Show that the number of simply connected closed paths through a 2xM rectangle grows at least as fast as M^{7}.
Some clarifications:
A path is simply connected if it does not contain any closed subpaths.
The 2xM rectangle encompass points [x,y] in the Cartesian plane with:
0 ≤ x ≤ 2 and 0 ≤ y ≤ M
Paths are constructed within the 2xM rectangle from vertical and horizontal segments connecting points with integral values in x and y.
An alternative statement of the problem is to show that the number of paths/M^{7} > 0 as M goes to infinity.
The boundary of the green area is just one such closed loop. 
