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In combinatorial game theory, the mex, or "minimum excludant", of a set of ordinals denotes the smallest ordinal not contained in the set. Some examples: where ω is the limit ordinal for the natural numbers. In the Sprague-Grundy theory the minimum excluded ordinal is used to determine the nimber of a normal-play impartial game, which is a game in which either player has the same moves in each position and the last player to move wins. The nimber is equal to 0 for a game that is lost immediately by the first player, and is equal to the mex of the nimbers of all possible next positions for any other game. For example, in a one-pile version of Nim, the game starts with a pile of n stones, and the player to move may take any positive number of stones. If n is zero, the nimber is 0. If n is 1, the player to move will leave 0 stones, and the mex of { 0 }, 1, gives the nimber for this case. If n is 2, the player to move can leave 0 or 1 stones, giving 2 as the mex of { 0, 1 }. In general, the player to move with a pile of n stones can leave anywhere from 0 to n-1 stones; the mex of the numbers { 0, 1, ..., n-1 } is always n. From this we can conclude that the first player wins if n is not zero.

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