Problem #12

On an infinite checkerboard, a game is played as follows. To begin, ab pieces are arranged on the board in an a×b block of adjoining squares, one piece per square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece that has been jumped over is then removed. For example, a legal sequence of moves is shown below in the 2×2 case

For what other values of a and b is it possible for the game to end with only one piece remaining on the board?

Source: International Mathematical Olympiad

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