MathNet 2015-part-03 — 191 problems (EN–VI) Question 8

Let $n \geqslant 2$. A piece is placed on each square of an $n \times n$ chessboard. A move consists in 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 it is possible that, after some number of moves, only $k$ squares contain at least one piece?