Question 55
Two players play a game on a $(m+1)$ by $m$ grid, which has $m+1$ horizontal lines, $m$ vertical lines, and hence $m(m+1)$ intersection points. One pebble is put on an intersection point. Two players alternatively move the pebble to an adjacent point along an edge, but they are not allowed to use edges used previously by either player. A player loses if he cannot move the pebble any more. Prove that there exists a winning strategy for the first player if the pebble was initially put on a point in the bottom horizontal line.