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Nimber
In mathematics, the nimbers, also called Grundy numbers (not to be confused with Grundy chromatic numbers), are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication.
Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering.
The nimber addition and multiplication operations are associative and commutative. Each nimber is its own additive inverse. In particular for some pairs of ordinals, their nimber sum is smaller than either addend. The minimum excludant operation is applied to sets of nimbers.
As a class, nimbers are indexed by ordinal numbers, and form a subclass of surreal numbers, introduced by John Horton Conway as part of his theory of combinatorial games. However, nimbers are distinct from ordinal and surreal numbers in that they follow distinct arithmetic rules, nim-addition and nim-multiplication. Other than that they are a proper class rather than a set, nimbers form a field under nim-addition and nim-multiplication.
As a set, finite nimbers can be put in one-to-one correspondence with finite ordinal numbers, which are the natural numbers. Nonetheless, their arithmetic structures are not isomorphic; nimber arithmetic fundamentally differs from ordinary arithmetic operations on natural numbers.
Nimbers are often denoted using a star notation {*0, *1, *2, ..., *ω, *(ω+1), ...}.
Nim is a game in which two players take turns removing objects from distinct heaps. As moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric, Nim is an impartial game. 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. The goal of the game is to be the player who removes the last object. The nimber of a heap is simply the number of objects in that heap. Using nim addition, one can calculate the nimber of the game as a whole. The winning strategy is to force the nimber of the game to 0 for the opponent's turn.
Cram is a game often played on a rectangular board in which players take turns placing dominoes either horizontally or vertically until no more dominoes can be placed. The first player that cannot make a move loses. As the possible moves for both players are the same, it is an impartial game and can have a nimber value. For example, any board that is an even size by an even size will have a nimber of 0. Any board that is even by odd will have a non-zero nimber. Any 2 × n board will have a nimber of 0 for all even n and a nimber of 1 for all odd n.
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Nimber
In mathematics, the nimbers, also called Grundy numbers (not to be confused with Grundy chromatic numbers), are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication.
Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering.
The nimber addition and multiplication operations are associative and commutative. Each nimber is its own additive inverse. In particular for some pairs of ordinals, their nimber sum is smaller than either addend. The minimum excludant operation is applied to sets of nimbers.
As a class, nimbers are indexed by ordinal numbers, and form a subclass of surreal numbers, introduced by John Horton Conway as part of his theory of combinatorial games. However, nimbers are distinct from ordinal and surreal numbers in that they follow distinct arithmetic rules, nim-addition and nim-multiplication. Other than that they are a proper class rather than a set, nimbers form a field under nim-addition and nim-multiplication.
As a set, finite nimbers can be put in one-to-one correspondence with finite ordinal numbers, which are the natural numbers. Nonetheless, their arithmetic structures are not isomorphic; nimber arithmetic fundamentally differs from ordinary arithmetic operations on natural numbers.
Nimbers are often denoted using a star notation {*0, *1, *2, ..., *ω, *(ω+1), ...}.
Nim is a game in which two players take turns removing objects from distinct heaps. As moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric, Nim is an impartial game. 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. The goal of the game is to be the player who removes the last object. The nimber of a heap is simply the number of objects in that heap. Using nim addition, one can calculate the nimber of the game as a whole. The winning strategy is to force the nimber of the game to 0 for the opponent's turn.
Cram is a game often played on a rectangular board in which players take turns placing dominoes either horizontally or vertically until no more dominoes can be placed. The first player that cannot make a move loses. As the possible moves for both players are the same, it is an impartial game and can have a nimber value. For example, any board that is an even size by an even size will have a nimber of 0. Any board that is even by odd will have a non-zero nimber. Any 2 × n board will have a nimber of 0 for all even n and a nimber of 1 for all odd n.