2002 Putnam A4

In Determinant Tic-Tac-Toe, Player 1 enters a 1 in an empty

$3 \times 3$ matrix. Player 0 counters with a 0 in a vacant position,

and play continues in turn until the $3 \times 3$ matrix is

completed with five 1's and four 0's. Player 0 wins if the

determinant is 0 and player 1 wins otherwise. Assuming both

players pursue optimal strategies, who will win and how?

在决定性井字棋中，玩家 1 在空的空格中输入 1

$3 \times 3$ 矩阵。玩家 0 在空位上用 0 反击，

依次进行，直到 $3 \times 3$ 矩阵为

由五个 1 和四个 0 组成。玩家 0 获胜，如果

行列式为 0，否则玩家 1 获胜。假设两者

玩家追求最优策略，谁会获胜以及如何获胜？