IMO 1982 LL POL40

Origin: POL

Problem

We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a 3k \times n rectangle, then it is impossible to reach a position with only one pawn on the board.