KMO 2009

There is a $m \times (m-1)$ board. (i.e. there are $m+1$ horizontal lines and $m$ vertical lines) A stone is put on an intersection of the lowest horizontal line. Now two players move this stone with the following rules. (i) Each players move the stone to a neighboring intersection along a segment, by turns. (ii) A segment, which is already passed by the stone, cannot be used more. (iii) One who cannot move the stone anymore loses. Prove that there is a winning strategy for the former player.

Find all pairs of two positive integers $(m,n)$ satisfying $3^m - 7^n = 2$ .