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36-592: Nim is a mathematical two player game. Nim or NIM may also refer to: Nim Nim is fundamental to the Sprague–Grundy theorem , which essentially says that every impartial game is equivalent to a nim game with a single pile. 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-shízi ( 捡石子 ), or "picking stones" —but

72-401: A misère game are the same until the number of heaps with at least two objects is exactly equal to one. At that point, the next player removes either all objects (or all but one) from the heap that has two or more, so no heaps will have more than one object (in other words, so all remaining heaps have exactly one object each), so the players are forced to alternate removing exactly one object until

108-414: A normal nim game, the player making the first move has a winning strategy if and only if the nim-sum of the sizes of the heaps is not zero. Otherwise, the second player has a winning strategy. Proof: Notice that the nim-sum (⊕) obeys the usual associative and commutative laws of addition (+) and also satisfies an additional property, x  ⊕  x  = 0. Let x 1 , ..., x n be

144-417: A number of objects are placed in an initial heap and two players alternately divide a heap into two nonempty heaps of different sizes. Thus, six objects may be divided into piles of 5+1 or 4+2, but not 3+3. Grundy's game can be played as either misère or normal play. Genus theory#Tame In the mathematical theory of games , genus theory in impartial games is a theory by which some games played under

180-406: A position s ≠ 0, and therefore this situation has to arise when it is the turn of the player following the winning strategy. The normal play strategy is for the player to reduce this to size 0 or 1, leaving an even number of heaps with size 1, and the misère strategy is to do the opposite. From that point on, all moves are forced. In another game which is commonly known as nim (but is better called

216-399: A position is exactly like a nim heap under the misère play convention, but classifying a game as nim means that it is equivalent to a nim heap. A game is a nim game, if: These are positions which we can pretend are nim positions (note difference between nim positions, which can be many nim heaps added together, and a single nim heap, which can only be 1 nim heap). A game G is tame if: Note

252-492: A single heap, one is permitted to remove the same number of objects from each heap. Yet another variation of nim is "circular nim", wherein any number of objects are placed in a circle and two players alternately remove one, two or three adjacent objects. For example, starting with a circle of ten objects, three objects are taken in the first move then another three then one but then three objects cannot be taken out in one move. In Grundy's game , another variation of nim,

288-518: 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

324-499: 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. The practical strategy to win at

360-547: Is equivalent to *0. It is important for further understanding of Genus theory, to know how reversible moves work. Suppose there are two games A and B, where A and B have the same options (moves available), then they are of course, equivalent. If B has an extra option, say to a game X, then A and B are still equivalent if there is a move from X to A. That is, B is the same as A in every way, except for an extra move (X), which can be reversed. Different games (positions) can be classified into several types: This does not mean that

396-548: Is played—and has symbolic importance—in the French New Wave film Last Year at Marienbad (1961). 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

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432-424: Is possible to make a move so that t = 0. Proof: Let d be the position of the leftmost (most significant) nonzero bit in the binary representation of s , and choose k such that the d th bit of x k is also nonzero. (Such a k must exist, since otherwise the d th bit of s would be 0.) Then letting y k  =  s  ⊕  x k , we claim that y k  <  x k : all bits to

468-505: Is the "100 game": Two players start from 0 and alternately add a number from 1 to 10 to the sum. The player who reaches 100 wins. The winning strategy is to reach a number in which the digits are subsequent (e.g., 01, 12, 23, 34,...) and control the game by jumping through all the numbers of this sequence. Once a player reaches 89, the opponent can only choose numbers from 90 to 99, and the next answer can in any case be 100. In another variation of nim, besides removing any number of objects from

504-451: 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

540-436: 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

576-408: The mex (minimum excludant) of the options of a game. g+ is the grundy value or nimber of a game under the normal play convention. g- or λ 0 is the outcome class of a game under the misère play convention. More specifically, to find g+, *0 is defined to have g+ = 0, and all other games have g+ equal to the mex of its options. To find g−, *0 has g− = 1, and all other games have g− equal to

612-460: The misère play convention can be analysed, to predict the outcome class of games. Genus theory was first published in the book On Numbers and Games , and later in Winning Ways for your Mathematical Plays Volume 2. Unlike the Sprague–Grundy theory for normal play impartial games, genus theory is not a complete theory for misère play impartial games. The genus of a game is defined using

648-423: The subtraction game ), an upper bound is imposed on the number of objects that can be removed in a turn. Instead of removing arbitrarily many objects, a player can only remove 1 or 2 or ... or k at a time. This game is commonly played in practice with only one heap. Bouton's analysis carries over easily to the general multiple-heap version of this game. The only difference is that as a first step, before computing

684-405: The example above, taking the nim-sum of the sizes is X = 3 ⊕ 4 ⊕ 5 = 2 . The nim-sums of the heap sizes A=3, B=4, and C=5 with X=2 are The only heap that is reduced is heap A, so the winning move is to reduce the size of heap A to 1 (by removing two objects). As a particular simple case, if there are only two heaps left, the strategy is to reduce the number of objects in the bigger heap to make

720-417: The game ends. In normal play, the player leaves an even number of non-zero heaps, so the same player takes last; in misère play, the player leaves an odd number of non-zero heaps, so the other player takes last. In a misère game with heaps of sizes three, four and five, the strategy would be applied like this: The soundness of the optimal strategy described above was demonstrated by C. Bouton. Theorem . In

756-414: 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. Normal-play nim (or more precisely the system of nimbers ) is fundamental to

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792-419: 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

828-491: The heaps equal. After that, no matter what move the opponent makes, the player can make the same move on the other heap, guaranteeing that they take the last object. When played as a misère game, nim strategy is different only when the normal play move would leave only heaps of size one. In that case, the correct move is to leave an odd number of heaps of size one (in normal play, the correct move would be to leave an even number of such heaps). These strategies for normal play and

864-458: The left of d are the same in x k and y k , bit d decreases from 1 to 0 (decreasing the value by 2 ), and any change in the remaining bits will amount to at most 2 −1. The first player can thus make a move by taking x k  −  y k objects from heap k , then The modification for misère play is demonstrated by noting that the modification first arises in a position that has only one heap of size 2 or more. Notice that in such

900-443: The length of the game from these two lemmas. Lemma 1 . If s = 0, then t ≠ 0 no matter what move is made. Proof: If there is no possible move, then the lemma is vacuously true (and the first player loses the normal play game by definition). Otherwise, any move in heap k will produce t  =  x k  ⊕  y k from (*). This number is nonzero, since x k  ≠  y k . Lemma 2 . If s ≠ 0, it

936-469: The mex of the g− of its options. λ 1 , λ 2 ..., is equal to the g− value of a game added to a number of *2 nim games, where the number is equal to the subscript. Thus the genus of a game is g . *0 has genus value 0 . Note that the superscript continues indefinitely, but in practice, a superscript is written with a finite number of digits, because it can be proven that eventually, the last 2 digits alternate indefinitely... It can be used to predict

972-407: The nim-sum is not zero before the move. If the nim-sum is zero, then the next player will lose if the other player does not make a mistake. To find out which move to make, let X be the nim-sum of all the heap sizes. Find a heap where the nim-sum of X and heap-size is less than the heap-size; the winning strategy is to play in such a heap, reducing that heap to the nim-sum of its original size with X. In

1008-671: The nim-sums we must reduce the sizes of the heaps modulo k  + 1. If this makes all the heaps of size zero (in misère play), the winning move is to take k objects from one of the heaps. In particular, in ideal play from a single heap of n objects, the second player can win if and only if This follows from calculating the nim-sequence of S (1, 2, ..., k ), 0.123 … k 0123 … k 0123 … = 0 ˙ .123 … k ˙ , {\displaystyle 0.123\ldots k0123\ldots k0123\ldots ={\dot {0}}.123\ldots {\dot {k}},} from which

1044-408: The ordinary sum, x + y . An example of the calculation with heaps of size 3, 4, and 5 is as follows: 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: In normal play, the winning strategy is to finish every move with a nim-sum of 0. This is always possible if

1080-456: 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

1116-444: The outcome of: In addition, some restive or restless pairs can form tame games, if they are equivalent. Two games are equivalent if they have the same options, where the same options are defined as options to equivalent games. Adding an option from which there is a reversible move does not affect equivalency. Some restive pairs, when added to another restive game of the same species, are still tame. A half tame game, added to itself,

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1152-1867: The sizes of the heaps before a move, and y 1 , ...,  y n the corresponding sizes after a move. Let s  =  x 1  ⊕ ... ⊕  x n and t  =  y 1  ⊕ ... ⊕  y n . If the move was in heap k , we have x i  =  y i for all i ≠ k , and x k  >  y k . By the properties of ⊕ mentioned above, we have t = 0 ⊕ t = s ⊕ s ⊕ t = s ⊕ ( x 1 ⊕ ⋯ ⊕ x n ) ⊕ ( y 1 ⊕ ⋯ ⊕ y n ) = s ⊕ ( x 1 ⊕ y 1 ) ⊕ ⋯ ⊕ ( x n ⊕ y n ) = s ⊕ 0 ⊕ ⋯ ⊕ 0 ⊕ ( x k ⊕ y k ) ⊕ 0 ⊕ ⋯ ⊕ 0 = s ⊕ x k ⊕ y k ( ∗ ) t = s ⊕ x k ⊕ y k {\displaystyle {\begin{aligned}t&=0\oplus t\\&=s\oplus s\oplus t\\&=s\oplus (x_{1}\oplus \cdots \oplus x_{n})\oplus (y_{1}\oplus \cdots \oplus y_{n})\\&=s\oplus (x_{1}\oplus y_{1})\oplus \cdots \oplus (x_{n}\oplus y_{n})\\&=s\oplus 0\oplus \cdots \oplus 0\oplus (x_{k}\oplus y_{k})\oplus 0\oplus \cdots \oplus 0\\&=s\oplus x_{k}\oplus y_{k}\\[10pt](*)\quad t&=s\oplus x_{k}\oplus y_{k}\end{aligned}}} The theorem follows by induction on

1188-409: The strategy above follows by the Sprague–Grundy theorem . The game "21" is played as a misère game with any number of players who take turns saying a number. The first player says "1" and each player in turn increases the number by 1, 2, or 3, but may not exceed 21; the player forced to say "21" loses. This can be modeled as a subtraction game with a heap of 21 − n objects. The winning strategy for

1224-432: 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

1260-436: The two-player version of this game is to always say a multiple of 4; it is then guaranteed that the other player will ultimately have to say 21; so in the standard version, wherein the first player opens with "1", they start with a losing move. The 21 game can also be played with different numbers, e.g., "Add at most 5; lose on 34". A sample game of 21 in which the second player follows the winning strategy: A similar version

1296-639: Was displayed at the Festival of Britain in 1951. In 1952, Herbert Koppel, Eugene Grant and Howard Bailer, 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

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