Consider the paths that join opposite corners of a rectangular grid and are not self-intersecting. Each path consists of single steps in one of eight directions, joining points with integral coordinates. This Demonstration shows the paths corresponding to grids of sides up to . A direction is indicated by the position of the selector around the word "directions" in the controls above. Any combination of directions can be chosen to be used by the steps forming the paths. The number of paths is shown, and the paths can be sequentially selected backward and forward.