2021 USAMO 第 3 题

Let $n \geq 2$ be an integer. An $n \times n$ board is initially empty. Each minute, you may perform one of three moves:

- If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells.
- If all cells in a column have a stone, you may remove all stones from that column.
- If all cells in a row have a stone, you may remove all stones from that row.

$$[asy] unitsize(20); draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)); fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey); draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2)); draw((0,2)--(4,2)); draw((2,4)--(2,0)); [/asy]$$

For which $n$ is it possible that, after some non-zero number of moves, the board has no stones?

设 $n \geq 2$ 为整数。 $n \times n$ 棋盘最初是空的。每分钟，您可以执行以下三个动作之一：

- 如果棋盘上有一个由三个单元格组成的 L 形特罗米诺区域，但没有棋子(见图；不允许旋转)，您可以在每个单元格中放置一颗棋子。
- 如果一列中的所有单元格都有石头，您可以移除该列中的所有石头。
- 如果一行中的所有单元格都有石头，您可以移除该行中的所有石头。

$$

[asy] 单位大小(20)；绘制((0,0)--(4,0)--(4,4)--(0,4)--(0,0))； fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--循环，灰色);绘制((0.2,3.8)--(1.8,3.8)--(1.8,1.8)--(3.8,1.8)--(3.8,0.2)--(0.2,0.2)--(0.2,3.8),线宽(2));绘制((0,2)--(4,2))；绘制((2,4)--(2,0))； [/asy]

$$

对于哪$n$，经过一些非零次数的移动后，棋盘上可能没有棋子？