IMO 1996 Problem 1

We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $AB = 20, BC = 12$ . The rectangle is divided into a grid of $20 \times 12$ 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 $\sqrt {r}$ . The task is to find a sequence of moves leading from the square with $A$ as a vertex to the square with $B$ as a vertex.

(a) Show that the task cannot be done if $r$ is divisible by 2 or 3.

(b) Prove that the task is possible when $r = 73$ .

(c) Can the task be done when $r = 97$ ?