2004 USAMO 第 4 题

Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing on the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from on to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.

Alice 和 Bob 在 6 x 6 网格上玩游戏。在他或她的回合中，玩家选择一个尚未出现在网格上的有理数，并将其写在网格的空白方块中。爱丽丝先走，然后玩家轮流走。当所有方格都写有数字时，在每一行中，数字最大的方格被涂成黑色。如果爱丽丝能够从网格顶部到网格底部画一条保持在黑色方块中的线，则她获胜；如果她不能，则鲍勃获胜。 (如果两个方格共享一个顶点，爱丽丝可以在这两个方格中从一个到另一个方格画一条线。)通过证明，找到其中一个玩家的获胜策略。