2018 CMO 第 5 题

Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$ .

Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.

令 $n \geq 3$ 为奇数，并假设 $n \times n$ 棋盘中的每个方格都是黑色或白色。如果两个方块具有相同颜色并共享一个公共顶点，则认为它们是相邻的；如果存在一系列方块 $c_1,\ldots,c_k$ 且 $c_1 = a, c_k = b$ 使得 $c_i, c_{i+1}$ 相邻(对于 $i=1,2,\ldots,k-1$ )，则认为两个方块 $a,b$ 是相连的。

找到最大数量 $M$，使得存在允许 $M$ 成对断开的方块的着色。