Aliquot Game

Aliquot part is another name for a proper divisor, i.e. any divisor of a given number other than the number itself. A prime number has only one aliquot part - the number 1. 1, 2, 3, 4, 6 are all aliquot parts of 12. The number 1 does not have aliquot parts. In the Aliquot game, players take turns subtracting an aliquot part of the number left by their opponent. The winner is the last player able to perform such a subtraction. The loser is the player left with a number that has no aliquot parts - 1. Thus the objective of the game is to leave your opponent without a move. What is the winning strategy?

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The game is extremely simple. The winning strategy is based on the observation that an odd number may only have odd divisors. It then follows that, in the game, an odd number is always followed by an even number. Obviously, the game may not end at an even number. A player that faces an even number can't be a loser on his/her turn. Such a player is assured of a move because any even number has at least one odd divisor. The player who manages to leave an odd number after every move is bound, therefore, to win the game. Why the game is simple? Because, 1 is as good an odd divisor as any other. If you face an odd number, you may only hope that your opponent will blunder on the next move. But faced with an even number, all it takes is to subtract 1.

All you need to play the Aliquot game is a piece of paper and a pencil. If you do not want to easily give the secret away, you should avoid subtracting 1 all the time. You may even introduce an additional rule that precludes subtracting 1 from a composite (non-prime) number .

(Note: if you dislike the starting number in the applet above you can modify it by clicking a little off its centerline.)

References

D. L. Silverman, Your Move, McGraw-Hill, 1971

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