MathDB

A token is placed in one square of a

m\times n

board, and is moved according to the following rules:
[*]In each turn, the token can be moved to a square sharing a side with the one currently occupied.
[*]The token cannot be placed in a square that has already been occupied.
[*]Any two consecutive moves cannot have the same direction.
The game ends when the token cannot be moved. Determine the values of

m

and

n

for which, by placing the token in some square, all the squares of the board will have been occupied in the end of the game.

Problem Statement