Question 126
Let $n$ be a positive integer. Find the smallest positive integer $k$ with the following property: there is a way to paint $k$ cells of a white $2n \times 2n$ board black such that there is exactly one way to cover the board with $1 \times 2$ and $2 \times 1$ dominoes satisfying:
(1) every domino lies along the grid lines of the board and does not stick out of the board;
(2) no two dominoes overlap;
(3) every domino covers at most $1$ black cell.
(1) every domino lies along the grid lines of the board and does not stick out of the board;
(2) no two dominoes overlap;
(3) every domino covers at most $1$ black cell.