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 same proper class as the ordinal numbers but 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.
Definition As a class, nimbers are indexed by ordinal numbers, and form a subclass of surreal numbers, introduced by John 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. The field of nimbers is denoted On2. 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 ) , . . . } {\displaystyle \{*0,*1,*2,...,*\omega ,*(\omega +1),...\}} . Alternatively, expressions that are interpreted as ordinary arithmetic are customarily enclosed in square brackets; for example, 2 2 = ∗ ( 2 ) 2 = 3 {\displaystyle 2^{2}=*(2)^{2}=3} in nim-multiplication, but [ 2 2 ] = ∗ ( 2 2 ) = 4 {\displaystyle [2^{2}]=*(2^{2})=4} .
Uses
Nim
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
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.
Northcott's game In Northcott's game, pegs for each player are placed along a column with a finite number of spaces. Each turn each player must move the piece up or down the column, but may not move past the other player's piece. Several columns are stacked together to add complexity. The player that can no longer make any moves loses. Unlike many other nimber related games, the number of spaces between the two tokens on each row are the sizes of the Nim heaps. If your opponent increases the number of spaces between two tokens, just decrease it on your next move. Else, play the game of Nim and make the Nim-sum of the number of spaces between the tokens on each row be 0.
Hackenbush
Hackenbush is a game invented by mathematician John Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line. Players take turns removing line segments. An impartial game version, thereby a game able to be analyzed using nimbers, can be found by removing distinction from the lines, allowing either player to cut any branch. Any segments reliant on the newly removed segment in order to connect to the ground line are removed as well. In this way, each connection to the ground can be considered a nim heap with a nimber value. Additionally, all the separate connections to the ground line can also be summed for a nimber of the game state.
Addition Nimber addition (also known as nim-addition) can be used to calculate the size of a single nim heap equivalent to a collection of nim heaps. It is defined recursively by
α ⊕ β = mex ( { α ′ ⊕ β : α ′ < α } ∪ { α ⊕ β ′ : β ′ < β } ) , {\displaystyle \alpha \oplus \beta =\operatorname {mex} \!{\bigl (}\{\alpha '\oplus \beta :\alpha '<\alpha \}\cup \{\alpha \oplus \beta ':\beta '<\beta \}{\bigr )},}
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