2012 IMO Shortlist C5

The columns and the rows of a $3 n \times 3 n$ square board are numbered $1,2, \ldots, 3 n$. Every square $(x, y)$ with $1 \leq x, y \leq 3 n$ is colored asparagus, byzantium or citrine according as the modulo 3 remainder of $x+y$ is 0,1 or 2 respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are $3 n^{2}$ tokens of each color. Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.

$3 n \times 3 n$ 方板的列和行编号为 $1,2, \ldots, 3 n$。每个带有 $1 \leq x, y \leq 3 n$ 的正方形 $(x, y)$ 根据 $x+y$ 的模 3 余数分别为 0,1 或 2，被着色为芦笋色、拜占庭色或黄水晶色。每个方块上放置一个芦笋色、拜占庭色或黄水晶色的标记，因此每种颜色有 $3 n^{2}$ 个标记。假设可以对标记进行排列，使得每个标记从其原始位置移动到最多 $d$ 的距离，每个芦笋标记替换一个拜占庭标记，每个拜占庭标记替换一个黄水晶标记，每个黄水晶标记替换一个芦笋标记。证明可以对标记进行排列，使得每个标记从其原始位置移动到最多 $d+2$ 的距离，并且每个方格包含一个与该方格颜色相同的标记。