2014 USAMO 第 4 题

Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, $B$ may choose any counter on the board and remove it. If at any time there are $k$ consecutive grid cells in a line all of which contain a counter, $A$ wins. Find the minimum value of $k$ for which $A$ cannot win in a finite number of moves, or prove that no such minimum value exists.

令 $k$ 为正整数。两个玩家 $A$ 和 $B$ 在无限的正六边形网格上玩游戏。最初所有网格单元都是空的。然后玩家轮流轮流，$A$ 首先移动。在他的行动中，$A$ 可以选择网格中两个相邻的空六边形，并在其中放置一个指示物。在他的行动中，$B$ 可以选择棋盘上的任何指示物并将其移除。如果在任何时候一行中有 $k$ 个连续的网格单元，并且所有网格单元都包含计数器，则 $A$ 获胜。找出 $k$ 的最小值，使得 $A$ 在有限步数中无法获胜，或者证明不存在这样的最小值。