Question 33
Let $n$ be a positive integer. Determine all integers $1 \le k \le 2n^2$ satisfying the following condition.
> There is a $2n \times 2n$ board. When $k$ distinct cells are chosen and colored black, while the other cells are colored white, the minimum number of the $2 \times 2$ squares which have both black and white cells is $2n-1$.
> There is a $2n \times 2n$ board. When $k$ distinct cells are chosen and colored black, while the other cells are colored white, the minimum number of the $2 \times 2$ squares which have both black and white cells is $2n-1$.