Suppose you're on a 4 × 4 grid and want to go from the bottom left to the top left. How many different paths can you take? You are only allowed to move up-down-left-right (not diagonally) and MUST pass through every square only once.
A thought: Let's say we move L, R, U, D. It must be L=R (since we end up in the same column) and also U-D=3, if this is of any help. Obviously the total number of moves must be 14. For example, LLLLLLRRRRRRUUU or RRRRLLLLUUUUDD in any order. Then we have to apply the restrictions, that we can't have, for example, RLR or LRL or UDU or DUD because this way we would pass through the same square twice.
I am stuck!