2288: Fliping game

Alice and Bob are playing a kind of special game on an N*M board (N rows, M columns). At the beginning, there are N*M coins in this board with one in each grid and every coin may be upward or downward freely. Then they take turns to choose a rectangle (x1, y1)-(n, m) (1 ≤ x1≤ n, 1 ≤ y1≤ m) and flips all the coins (upward to downward, downward to upward) in it (i.e. flip all positions (x, y) where x1≤ x ≤ n, y1≤ y ≤ m)). The only restriction is that the top-left corner (i.e. (x1, y1)) must be changing from upward to downward. The game ends when all coins are downward, and the one who cannot play in his (her) turns loses the game. Here's the problem: Who will win the game if both use the best strategy? You can assume that Alice always goes first.

For each case, output the winner’s name, either Alice or Bob.

2
2 2
1 1
1 1
3 3
0 0 0
0 0 0
0 0 0

Alice
Bob