Question 69
For $n \geq 3$, it is given an $2n \times 2n$ board with black and white squares. It is known that all border squares are black and no $2 \times 2$ subboard has all four squares of the same color. Prove that there exists a $2 \times 2$ subboard painted like a chessboard, i.e. with two opposite black corners and two opposite white corners.