There are several pebbles placed on an n × n chessboard, such that each pebble is inside a square and no two pebbles share the same square.
Perry decides to play the following game. At each turn, he moves one of the pebbles to an empty neighboring square. After a while, Perry notices that every pebble has passed through every square of the chessboard exactly once and has come back to its original position.
Prove that there was a moment when no pebble was on its original position.