Name a number no one's named before — but never be the one who says 1
Two players alternately call out a positive whole number that cannot be written as a sum of numbers already named (a number may be reused any number of times in that sum). For example, once 3 has been named, no one may later name 6 or 9. Whoever is forced to name the number 1 loses, since every remaining number can then be built from it.
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The surprising part isn't who wins — that stays unsolved for most starting positions — it's that the game is guaranteed to end at all. A result related to the classic 'coin problem' in number theory shows that once enough numbers with no common factor have been named, every sufficiently large number becomes unavailable, so someone is eventually forced into naming 1, even though nothing about the rules makes that obvious in advance. Devised by John Conway and detailed in Berlekamp, Conway and Guy's Winning Ways for Your Mathematical Plays (1982).
— Elwyn Berlekamp, John Conway, Richard Guy, Winning Ways for Your Mathematical Plays — 1982
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