one-player game on an infinite chessboard
Source: 1983 Polish MO Finals p3
February 25, 2020
Chessboardinfinite chessboardcombinatorics
Problem Statement
Consider the following one-player game on an infinite chessboard. If two horizontally or vertically adjacent squares are occupied by a pawn each, and a square on the same line that is adjacent to one of them is empty, then it is allowed to remove the two pawns and place a pawn on the third (empty) square. Prove that if in the initial position all the pawns were forming a rectangle with the number of squares divisible by , then it is not possible to end the game with only one pawn left on the board.