The value of a game position, where mex is the smallest nonnegative integer missing from the values of its legal successors. Terminal positions have value zero. The Sprague-Grundy theorem explains why these values combine by exclusive or.
For finite nimbers, multiplication is recursively defined from smaller arguments bywhere is Nim-sum and mex selects the smallest missing nonnegative integer. Thus zero times anything is zero and one times is . For , the displayed set is , so .
Together with Nim-sum, this is field arithmetic on finite nimbers, rather than ordinary integer multiplication. In particular cannot be computed by a usual integer product or bitwise exclusive or.[ref]
https://circles.math.ucla.edu/circles/lib/data/Handout-4717-4806.pdf. Aaron Anderson, Nimbers, 3 October 2021, multiplication definition
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The Grundy number, also known as the nimber, is a concept from combinatorial game theory used to analyze games, particularly impartial games. It is a measure of a position's winning potential in these games. In an impartial game, the players have the same options available to them regardless of who is about to move. A position in such a game can have a Grundy number that helps determine whether it is a winning position (for the player about to move) or a losing position.