Let $n\geq 2$ be a positive integer. On an $n\times n$ board, $n$ rooks are placed in such a manner that no two attack each other. All rooks move at the same time and are only allowed to move in a square adjacent to the one in which they are located. Determine all the values of $n$ for which there is a placement of the rooks so that, after a move, the rooks still do not attack each other. Note: Two squares are adjacent if they share a common side.