Nim

Nim is a two-player game in which players take objects from one or more heaps, usually removing any positive number from one heap per turn. Under normal play, the player who takes the last object wins.

Nim-sum: The bitwise XOR of a collection of nonnegative integers, used to characterize positions in impartial games. A position’s Nim-sum tells whether it is winning under normal play.
Bitwise XOR: A binary operation that returns 1 for each bit position where its inputs differ. Nim-sum is repeated XOR applied to the heap sizes.
Charles L. Bouton: An American mathematician who published a mathematical analysis of Nim in 1901. His analysis established the XOR criterion for ordinary Nim.
Kayles: An impartial game in which players remove one or two adjacent pins from a row. Its positions can be analyzed by assigning nimbers to independent segments.
Sprague–Grundy theorem: A theorem that assigns impartial positions nimbers whose XOR represents their combined game. It extends Nim’s heap arithmetic to sums of many impartial games.
Binary numeral system: A positional numeral system using the digits 0 and 1. Nim’s winning move is found by comparing heap sizes in binary.
The Game of Nim: Charles L. Bouton’s 1901 paper presenting a mathematical analysis of Nim. This paper gave the classic proof of Nim’s winning strategy.
Dawson's Kayles: An impartial game in which players remove two adjacent pins from a row. Its endgame positions illustrate the broader use of Nim-value analysis.
Winning position: A game position from which the player to move can force a win against perfect play. Nim’s optimal strategy is to move from a losing position to one of these.
Impartial game: A game in which the available moves from a position do not depend on which player is moving. Nim’s move rules are identical for both players.

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