Chessboard Diagonal Moves Minimum Occupied

Let $n \geqslant 2$. A piece is placed on each square of an $n \times n$ chessboard. A move consists of moving each piece to a square that touches its starting square at exactly one corner (several pieces may end up on the same square). What is the smallest integer $k$ such that, after some number of moves, it is possible that only $k$ squares contain at least one piece?