The Secret of Nim

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Explore the mathematical background of the game of Nim and learn how to play the optimal strategy.

Contributed by: Ralf Becker (March 2011)
Open content licensed under CC BY-NC-SA



The crucial factor for an optimal (winning) strategy is the nim-value. For details of how to calculate it, see Nim-Value (Wolfram MathWorld). The circle-plus symbol used in this Demonstration indicates the operation of calculating the nim-value.

Snapshot 1: if at any point in the game, the nim-value is zero for a given player, the position is safe, which means that player can win with correct play

Snapshot 2: otherwise, the position is unsafe and the player will always lose if the other player plays correctly

Snapshot 3: to achieve a safe position, find a heap whose size is less than the min-value of with the nim-value of all the heaps; reduce that heap to , resulting in a nim-value of zero

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