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An impartial game is a game in which both players have the same legal moves from any given position. Under the normal-play convention, a player with no legal move loses. For finite games without draws, Grundy numbers encode positions for the Sprague-Grundy theorem.

Every finite acyclic impartial game under the normal-play convention has a Grundy number. The Grundy number of a sum of independent games is the bitwise exclusive or of their individual Grundy numbers, so the sum is losing exactly when that value is zero.

Sprague-Grundy theory studies finite acyclic impartial games under the normal-play convention by assigning each position a Grundy number. The assignment recursively takes the minimum excludant of the values of all legal followers. The Sprague-Grundy theorem then reduces a disjunctive sum of games to the bitwise exclusive or of its component values. This framework explains both why a zero value is losing and how independently understood games can be combined without exploring the entire product of their state spaces.

For arrays indexed by $b$-bit nonnegative integers, their XOR convolution is $(a{\ast }_{\oplus }c{)}_r={\sum }_i{a}_i{c}_{i\oplus r}$, where $\oplus$ is bitwise exclusive or. If entries count choices of Grundy numbers, its repeated application counts sums of impartial games with a specified combined Grundy number. The Hadamard transform turns this operation into pointwise multiplication.