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. 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. Only the last move changes between misere and normal play.įor the generalisations, n and m can be any value > 0, and they may be the same. 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. ( In misère play he would take 2 from C leaving (0, 1, 0).) 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.īob takes 1 from C leaving two 1s. In misère play, the goal is instead to ensure that the opponent is forced to take the last remaining object. The goal is to be the last to take an object. The two players alternate taking any number of objects from any one of the heaps. The normal game is between two players and is played with three heaps of any number of objects. A version of Nim is played-and has symbolic importance-in the French New Wave film Last Year at Marienbad (1961). The game of Nim was the subject of Martin Gardner's February 1958 Mathematical Games column in Scientific American. A Nim Playing Machine has been described made from TinkerToy. Maxon Corporation, developed a machine weighing 23 kilograms (50 lb) which played Nim against a human opponent and regularly won. In 1952 Herbert Koppel, Eugene Grant and Howard Bailer, engineers from the W. Ferranti built a Nim playing computer which was displayed at the Festival of Britain in 1951. It was also one of the first-ever electronic computerized games. From May 11, 1940, to October 27, 1940, only a few people were able to beat the machine in that six-week period if they did, they were presented with a coin that said Nim Champ. Īt the 1940 New York World's Fair Westinghouse displayed a machine, the Nimatron, that played Nim. 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 is a special case of a poset game where the poset consists of disjoint chains (the heaps). Only tame games can be played using the same strategy as misère Nim. While all normal play impartial games can be assigned a Nim value, that is not the case under the misère convention. 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). 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. 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. In either normal play or a misère game, when the number of heaps with at least two objects is exactly equal to one, the player who takes next can easily win. This is called normal play because the last move is a winning move in most games, even though it is not the normal way that Nim is played. Nim can also be played as a normal play game whereby the player taking the last object wins. Nim is typically played as a misère game, in which the player to take the last object loses. 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. Its current name was coined by Charles L. The game is said to have originated in China-it closely resembles the Chinese game of 捡石子 jiǎn-shízi, or "picking stones" -but the origin is uncertain the earliest European references to Nim are from the beginning of the 16th century. Variants of Nim have been played since ancient times. 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. 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. Nim is a mathematical game of strategy in which two players take turns removing (or "nimming") objects from distinct heaps or piles. Players take turns to choose a row and remove any number of matches from it. Matches set up in rows for a game of Nim.
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