A common puzzle problem is to count the number of paths that start from the bottom-left-hand corner of a grid and end at the top-right hand corner, with the restriction that you can only move upwards or rightwards.
For example, for a 3 by 3 grid (as shown below), the total number of ways is $\binom{6}{3}=20$
If we modify the problem to allow the path to contain downwards and leftwards moves, but maintain a restriction that you cannot visit an intersection twice, what is a general solution for the number of possible paths?
(In the case of the 3 by 3 grid below, I enumerated the total paths with a computer program and got 210).
