I have a $N\times N$ chessboard, and I need to compute all the paths from a corner to the opposite, walking once on each tile, and only walking from a tile to one of the nearest tiles (i.e., no jumps, no diagonal moving).
I was thinking to solve it by recursion, like on a $1\times i$ chessboard I have one path for any $i$, and for a $2\times i$ I have the same number of paths than for a $2\times i-2$, and so on. But adding more rows is making it difficult, and I am getting swamped in too specific cases.
Is there a general problem behind this that I am not aware of? I cannot understand if it is a problem of combinatorics or of graph theor. I am ignorant in both of them ..
It is maybe related to domino covering problems, with the constraint that the tiling should form a path, maybe?