Say the maze consists of $M\times N$ cells and the visitor may enter a cell from another cell which shares a common side. The visitor starts from the top left cell and ends at the bottom right cell, visiting each cell exactly once. This is possible when $M,N$ are not both even. Question is, how many routes are there? Anybody has considered this problem before?
More generally, if the visitor starts from a given cell and ends at another given cell for which a qualifying route exists, then how many such routes are there?