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In mathematics, the nimbers, also called Grundy 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.[1] The minimum excludant operation is applied to sets of nimbers.
^Advances in computer games : 14th International Conference, ACG 2015, Leiden, the Netherlands, July 1-3, 2015, Revised selected papers. Herik, Jaap van den,, Plaat, Aske,, Kosters, Walter. Cham. 2015-12-24. ISBN 978-3319279923. OCLC 933627646.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link)
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...
game under the normal play convention is equivalent to a nimber. The representation of this nimber can change from game to game, but every possible state...
military decorations State decoration Army-Navy "E" Award Medal inflation "Nimber 1348.33, Volume 3" (PDF). Defense Technical Information Center. United States...
operations, there are also the "natural" arithmetic of ordinals and the nimber operations. The union of two disjoint well-ordered sets S and T can be well-ordered...
expense of continuity. Interpreted as nimbers, a game-theoretic variant of numbers, ordinals can also be combined via nimber arithmetic operations. These operations...
position in a partisan game can have a nimber as its value, or else the game would be impartial. However, some nimbers can still occur as the values of game...
Nimber addition is the exclusive or of nonnegative integers in binary representation. This is also the vector addition in ( Z / 2 Z ) 4 {\displaystyle...
way are known as nimbers. The Sprague–Grundy theorem states that every impartial game is equivalent to a nimber. The "smallest" nimbers – the simplest and...
maximum number of colors obtainable by a greedy graph coloring algorithm Nimber, a type of value used in combinatorial game theory, also called a Grundy...
asserts that every such game has an equivalent Sprague–Grundy value, a "nimber", which indicates the number of pieces in an equivalent position in the...
they may be the same. Normal-play nim (or more precisely the system of nimbers) is fundamental to the Sprague–Grundy theorem, which essentially says that...
{\displaystyle n} . This is often denoted K n {\displaystyle K_{n}} ; it is a nimber, not a number. By the Sprague–Grundy theorem, K n {\displaystyle K_{n}}...
Multiplicative group of integers modulo n).[citation needed] In a connection with nimbers, these numbers are often called Fermat 2-powers. The numbers 2 2 n {\displaystyle...
Berlekamp, Conway & Guy (2001), "Green hackenbush, the game of nim, and nimbers", pp. 40–42. Golomb (1966); Eppstein (2018) Whinihan (1963); Larsson &...
additional games whose values are not real numbers, such as star and all other nimbers. Further analysis of the game can be made using graph theory by considering...
recasting its conventional nimber-based solution into a multiplicative form, and then modifying that slightly for misere play. The nimbers that occur in the normal...
(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 λ0 is the outcome class...
Magazine 2, September 2014, pp. 41–52 (with Carlos Santos and Pedro Duarte) Nimbers in Partizan Games, Games of No Chance 4, R.J. Nowakowski (Ed.), MSRI Publications...
theory) Zero game, fuzzy game Dots and boxes Impartial game Digital sum Nim Nimber Sprague–Grundy theorem Partizan game Solved board games Col game Sim (pencil...
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