Consider a board game played on an m × n matrix. Player 1 has an unlimited supply of white chips, and player 2 has an unlimited supply of black chips. Starting with player 1, the players take turns claiming cells of the matrix. A player claims a cell by placing one of her chips in this cell. Once a cell is claimed, it cannot be altered. Players must claim exactly one cell in each round. The game ends after mn rounds, when all of the cells are claimed. At the end of the game, each cell is evaluated as either a "victory cell" or a "loss cell." A cell is classified as a victory if it shares sides with at least two cells of the same color. That is, there are at least two cells of the same color that are immediately left, right, up, or down from (not diagonal to) the cell being evaluated. A player gets one point for each of the victory cells that she claimed and for each of the loss cells that her opponent claimed. The player with the most points wins the game; if the players have the same number of points, then a tie is declared.
(a) Under what conditions on m and n do you know that one of the players has a strategy that guarantees a win? Can you determine which player can guarantee a win? If so, provide some logic or a proof.
(b) Repeat the analysis for a version of this game in which a victory cell must share sides with at least three cells of the same color.