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?

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Source: International Mathematical Olympiad
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