Nim

Nim is a mathematical combinatorial game in which two players take turns removing (or "nimming") objects from distinct heaps or piles. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap or pile. Depending on the version being played, the goal of the game is either to avoid taking the last object or to take the last object. Nim is fundamental to the Sprague–Grundy theorem, which essentially says that every impartial game is equivalent (when regarded as a subgame of a larger impartial game) to a nim game with a single pile.

History Variants of nim have been played since ancient times. The game is said to have originated in China—it closely resembles the Chinese game of jiǎn-shízǐ (捡石子), or "picking stones"—but the origin is uncertain; the earliest European references to nim are from the beginning of the 16th century. Its current name was coined by Charles L. Bouton of Harvard University, who also developed the complete theory of the game in 1901, but the origins of the name were never fully explained. The Oxford English Dictionary derives the name from the German verb nimm, meaning "take". At the 1939 New York World's Fair, Westinghouse displayed a machine, the Nimatron, that played nim. From May 11 to October 27, 1940, only a few people were able to beat the machine in that six-month period; if they did, they were presented with a coin that said "Nim Champ". It was also one of the first-ever electronic computerized games. Ferranti built a nim-playing computer which was displayed at the Festival of Britain in 1951. In 1952, Herbert Koppel, Eugene Grant and Howard Baller, engineers from the W. L. Maxson Corporation, developed a machine weighing 23 kilograms (50 lb) which played nim against a human opponent and regularly won. A nim playing machine has been described made from tinkertoys. The game of nim was the subject of Martin Gardner's February 1958 "Mathematical Games" column in Scientific American. A version of nim is played—and has symbolic importance—in the French New Wave film Last Year at Marienbad (1961).

Game play and illustration Nim is typically played as a misère game, in which the player to take the last object loses. Nim can also be played as a "normal play" game whereby the player taking the last object wins. In either normal play or a misère game, when there is exactly one heap with at least two objects, the player who takes next can easily win. If this removes either all or all but one objects from the heap that has two or more, then no heaps will have more than one object, so the players are forced to alternate removing exactly one object until the game ends. If the player leaves an even number of non-zero heaps (as the player would do in normal play), the player takes last; if the player leaves an odd number of heaps (as the player would do in misère play), then the other player takes last. The normal game is between two players and is played with three heaps of any number of objects. The two players alternate taking any number of objects from any one of the heaps. The goal is to be the last to take an object. In misère play, the goal is instead to ensure that the opponent is forced to take the last remaining object. The following example of a normal game is played between fictional players Bob and Alice, who start with heaps of three, four and five objects.

Winning positions The practical strategy to win at the game of nim is for a player to get the other into one of the following positions, and every successive turn afterwards they should be able to make one of the smaller positions. Only the last move changes between misère and normal play.

For the generalisations, n and m can be any value > 0, and they may be the same.

Mathematical theory Normal-play nim (or more precisely the system of nimbers) is fundamental to the Sprague–Grundy theorem, which essentially says that in normal play every impartial game is equivalent to a nim heap that yields the same outcome when played in parallel with other normal play impartial games (see disjunctive sum). While all normal-play impartial games can be assigned a nim value, that is not the case under the misère convention. Only tame games can be played using the same strategy as misère nim. Nim is a special case of a poset game where the poset consists of disjoint chains (the heaps). The evolution graph of the game of nim with three heaps is the same as three branches of the evolution graph of the Ulam–Warburton automaton. Nim has been mathematically solved for any number of initial heaps and objects, and there is an easily calculated way to determine which player will win and which winning moves are open to that player. The key to the theory of the game is the binary digital sum of the heap sizes, i.e., the sum (in binary), neglecting all carries from one digit to another. This operation is also known as "bitwise xor" or "vector addition over GF(2)" (bitwise addition modulo 2). Within combinatorial game theory it is usually called the nim-sum, as it will be called here. The nim-sum of x and y is written x ⊕ y to distinguish it from the ordinary sum, x + y. An example of the calculation with heaps of size 3, 4, and 5 is as follows:

Binary Decimal 0112 310 Heap A 1002 410 Heap B 1012 510 Heap C --- 0102 210 The nim-sum of heaps A, B, and C, 3 ⊕ 4 ⊕ 5 = 2

An equivalent procedure, which is often easier to perform mentally, is to express the heap sizes as sums of distinct powers of 2, cancel pairs of equal powers, and then add what is left:

3 = 0 + 2 + 1 = 2 1 Heap A 4 = 4 + 0 + 0 = 4 Heap B 5 = 4 + 0 + 1 = 4 1 Heap C -------------------------------------------------------------------- 2 = 2 What is left after canceling 1s and 4s

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