We are given a positive integer and a rectangular board with dimensions . The rectangle is divided into a grid of unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is . The task is to find a sequence of moves leading from the square with as a vertex to the square with as a vertex.
(a) Show that the task cannot be done if is divisible by 2 or 3.
(b) Prove that the task is possible when .
(c) Can the task be done when ?
(a) Show that the task cannot be done if is divisible by 2 or 3.
(b) Prove that the task is possible when .
(c) Can the task be done when ?