2024 IMO Shortlist C8

Let $n$ be a positive integer. Given an $n\times n$ board, the unit cell in the top left corner is initially coloured black, and the other cells are coloured white. We then apply a series of colouring operations to the board. In each operation, we choose a $2\times 2$ square with exactly one cell coloured black and we colour the remaining three cells of that $2\times 2$ square black.

Determine all values of $n$ such that we can colour the whole board black.

设 $n$ 为正整数。给定一个 $n\times n$ 棋盘，左上角的单位格初始为黑色，其余格子为白色。接着对棋盘进行一系列染色操作。每次操作选择一个恰有一个黑格的 $2\times 2$ 方块，并把这个 $2\times 2$ 方块中的另外三个格子染成黑色。

确定所有 $n$，使得可以把整个棋盘染成黑色。