Nimber

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In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication.

Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering.

The nimber addition and multiplication operations are associative and commutative. Each nimber is its own additive inverse. In particular for some pairs of ordinals, their nimber sum is smaller than either addend.^[1] The minimum excludant operation is applied to sets of nimbers.