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nimber
In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of nim heaps. They arise in a much larger class of games because of the Sprague–Grundy theorem. The nimbers are the ordinal numbers endowed with a new nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication.
==Properties==
The Sprague–Grundy theorem states that every impartial game is equivalent to a nim heap of a certain size. Nimber addition (also known as nim-addition) can be used to calculate the size of a single heap equivalent to a collection of heaps. It is defined recursively by
:
where for a set ''S'' of ordinals, mex(''S'') is defined to be the "minimum excluded ordinal", i.e. mex(''S'') is the smallest ordinal which is not an element of ''S''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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