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Inmathematics, thenimbers, also calledGrundy numbers (not to be confused withGrundy chromatic numbers), are introduced incombinatorial game theory, where they are defined as the values of heaps in the gameNim. The nimbers are theordinal numbers endowed withnimber addition andnimber multiplication, which are distinct fromordinal addition andordinal multiplication.

Because of theSprague–Grundy theorem which states that everyimpartial 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 inpartisan games likeDomineering.

The nimber addition and multiplication operations are associative and commutative. Each nimber is its ownadditive inverse. In particular for some pairs of ordinals, their nimber sum is smaller than either addend.^[1] Theminimum excludant operation is applied to sets of nimbers.

As a class,nimbers are indexed byordinal numbers, and form a subclass ofsurreal numbers, introduced byJohn Horton Conway as part of his theory ofcombinatorial games.^[2] However, nimbers are distinct from ordinal and surreal numbers in that they follow distinctarithmetic rules, nim-addition and nim-multiplication. Other than that they are a proper class rather than a set, nimbers form afield under nim-addition and nim-multiplication.

As a set, finite nimbers can be putin one-to-one correspondence with finite ordinal numbers, which are thenatural numbers. Nonetheless, their arithmetic structures are notisomorphic; 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.^[3]

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. Any2 ×n board will have a nimber of 0 for all evenn and a nimber of 1 for all oddn.

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.^[4]

Hackenbush is a game invented by mathematicianJohn Horton Conway. It may be played on any configuration of coloredline 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.

Nimber addition (also known asnim-addition) can be used to calculate the size of a single nim heap equivalent to a collection of nim heaps. It is defined recursively bywhere theminimum excludantmex(S) of a setS of ordinals is defined to be the smallest ordinal that isnot an element ofS.

For finite ordinals, thenim-sum is easily evaluated on a computer by taking thebitwiseexclusive or (XOR, denoted by⊕) of the corresponding numbers. For example, the nim-sum of 7 and 14 can be found by writing 7 as 111 and 14 as 1110; the ones place adds to 1; the twos place adds to 2, which we replace with 0; the fours place adds to 2, which we replace with 0; the eights place adds to 1. So the nim-sum is written in binary as 1001, or in decimal as 9.

This property of addition follows from the fact that bothmex and XOR yield a winning strategy for Nim and there can be only one such strategy; or it can be shown directly by induction: Letα andβ be two finite ordinals, and assume that the nim-sum of all pairs with one of them reduced is already defined. The only number whose XOR withα isα ⊕β isβ, and vice versa; thusα ⊕β is excluded.$${\displaystyle \zeta :=\alpha \oplus \beta \oplus \gamma }$$ On the other hand, for any ordinalγ <α ⊕β, XORingζ with all ofα,β andγ must lead to a reduction for one of them (since the leading 1 inζ must be present in at least one of the three); since$${\displaystyle \zeta \oplus \gamma =\alpha \oplus \beta >\gamma ,}$$ we must have eitherThusγ is included as either$${\displaystyle \begin{array}{r}(\beta \oplus \gamma )\oplus \beta ,\text{or}\\ \alpha \oplus (\alpha \oplus \gamma );\end{array}}$$and henceα ⊕β is the minimum excluded ordinal.

Nimber addition isassociative andcommutative, with0 as the additiveidentity element. Moreover, a nimber is its ownadditive inverse.^[5] It follows thatα ⊕β = 0if and only ifα =β.

Nimber multiplication (nim-multiplication) is defined recursively by

Nimber multiplication is associative and commutative, with the ordinal1 as the multiplicativeidentity element. Moreover, nimber multiplicationdistributes over nimber addition.^[5][a]

Thus, except for the fact that nimbers form aproper class and not a set, the class of nimbers forms aring. In fact, it even determines analgebraically closed field ofcharacteristic 2, with the nimber multiplicative inverse of a nonzero ordinalα given by

$${\displaystyle {\alpha }^{-1}=\mathrm{mex}(S),}$$whereS is the smallest set of ordinals (nimbers) such that

1. 0 is an element ofS;
2. if0 <α′ <α andβ' is an element ofS, then${\displaystyle (1\oplus ({\alpha }^{\prime }\oplus \alpha )\otimes {\beta }^{\prime })\otimes ({\alpha }^{\prime }{)}^{-1}}$ is also an element ofS.

For all natural numbersn, the set of nimbers less than2²ⁿ form theGalois fieldGF(2²ⁿ) of order 2²ⁿ. Therefore, the set of finite nimbers is isomorphic to thedirect limit asn → ∞ of the fieldsGF(2²ⁿ). This subfield is not algebraically closed, since no fieldGF(2^k) withk not a power of 2 is contained in any of those fields, and therefore not in their direct limit; for instance the polynomialx³ +x + 1, which has a root inGF(2³), does not have a root in the set of finite nimbers.

Just as in the case of nimber addition, there is a means of computing the nimber product of finite ordinals. This is determined by the rules that

1. The nimber product of a Fermat 2-power (numbers of the form2²ⁿ) with a smaller number is equal to their ordinary product;
2. The nimber square of a Fermat 2-powerx is equal to3x/2 as evaluated under the ordinary multiplication of natural numbers.

The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinalω^ωω, whereω is the smallest infinite ordinal. It follows that as a nimber,ω^ωω istranscendental over the field.^[6]

The following tables exhibit addition and multiplication among the first 16 nimbers.

This subset is closed under both operations, since 16 is of the form 2²ⁿ.(If you prefer simple text tables, they arehere.)

• Surreal number

• Combinatorial game theory
• Finite fields
• Ordinal numbers

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