nimmultiplication

Nimmultiplication, or nimber multiplication, is a binary operation defined on nimbers, the abstract values that arise in impartial combinatorial game theory through the Sprague-Grundy theorem. Nimbers encode the equivalence classes of impartial games under disjunctive sum, and they form a rich arithmetic with two main operations: nim-addition (bitwise XOR) and nim-multiplication (a more intricate product).

The multiplication of nimbers is defined recursively in a way that ensures consistency with Grundy theory and with nim-addition. Roughly speaking, to determine the product of two nimbers a and b, one looks at the options of a and b (games reachable in one move) and uses a mex-based rule to assign a new nimber as their product. The exact recursive definition is technical, but the essential idea is to produce an operation that makes nimbers closed under multiplication and that distributes over nim-addition.

Key properties include that nim-multiplication is commutative and associative, and it distributes over nim-addition. The zero nimber acts as an absorbing element, and there is a multiplicative identity. Unlike ordinary multiplication, nim-multiplication does not correlate simply with raw size or magnitude of the nimbers; its values reflect the structure of the underlying game options rather than numerical ordering.

Applications of nim-multiplication occur primarily in the analysis of complex impartial games and in theoretical explorations of nimber arithmetic. It provides a tool for computing the Grundy numbers of disjunctive sums and for understanding the algebraic structure of nimbers. Related concepts include nim-addition, Grundy numbers, and the broader study of impartial games.

See also: Nim-sum, Grundy numbers, impartial games, Conway’s On Numbers and Games.