Nimber

In mathematics, nimbers, also called Grundy numbers, are important ideas used in a branch called combinatorial game theory. They help describe the values of heaps in a fun game called Nim. Nimbers share some similarities with ordinal numbers, but they have their own special ways of adding and multiplying, known as nimber addition and nimber multiplication.

Thanks to something called the Sprague–Grundy theorem, nimbers are useful in many types of games where players take turns and there are no advantages for either player at the start — these are called impartial games. Nimbers can even show up in games where players have different powers, like Domineering.

Nimber addition and multiplication follow certain rules, like being associative and commutative. A special rule called the minimum excludant is used with sets of nimbers to find new values. These ideas help game theorists understand complex strategies and solve puzzles.

Definition

Nimbers are special numbers used in a game called Nim and in the study of combinatorial games. They were introduced by John Conway and are different from regular numbers because they follow unique rules for addition and multiplication. Even though nimbers can be matched with natural numbers, the way they add and multiply is not the same as with everyday math.

Nimbers are written using a star, like ⁎0, ⁎1, ⁎2, and so on, showing their special role in game theory.

Uses

Nim

Main article: Nim

Nim is a game where two players take turns removing objects from different heaps. The goal is to be the player who takes the last object. The nimber of a heap is simply the number of objects in it. By using nim addition, players can find a winning strategy.

Cram

Main article: Cram (game)

Cram is a game played on a rectangular board where players place dominoes either horizontally or vertically. The first player who cannot make a move loses. Like Nim, Cram is an impartial game and can have a nimber value. For example, boards that are even-sized by even-sized will have a nimber of 0.

Northcott's game

In Northcott's game, players move pegs up or down a column without passing each other’s peg. Several columns add complexity. The player who cannot make a move loses. The spaces between pegs act like Nim heaps, and players can use Nim strategies to win.

Hackenbush

Main article: Hackenbush

Hackenbush is a game invented by mathematician John Conway where players remove colored line segments connected to a ground line. In its impartial version, either player can cut any branch, and any segments that depend on the removed branch also fall off. Each connection to the ground can be thought of as a Nim heap with a nimber value.

Addition

Nimber addition, also called nim-addition, helps us find the size of a single Nim heap that is the same as a group of Nim heaps together. It is a special way of adding numbers that works for Nim games.

Nimber addition has some nice properties: it is associative (we can change the order of addition without changing the result) and commutative (the order of the numbers does not matter). The number 0 is the identity element, meaning adding 0 does not change the nimber. Also, every nimber is its own opposite, so adding a nimber to itself gives 0. This means that α ⊕ β equals 0 only when α and β are the same.

Multiplication

Nimber multiplication, also called nim-multiplication, follows special rules that are different from normal multiplication. One important rule is that multiplying a Fermat 2-power, like 2²ⁿ, by a smaller number gives the same result as regular multiplication. Another rule says that the nimber square of a Fermat 2-power equals 3·2²ⁿ⁻¹ when using regular multiplication.

These rules help us understand how nimbers behave in calculations. The set of all finite nimbers connects to mathematical structures called fields, similar to systems used in advanced algebra.

Addition and multiplication tables

The following tables show addition and multiplication for the first 16 nimbers, which are special numbers used in a game called Nim. These numbers follow special rules for adding and multiplying that are different from the usual rules you might learn in school. Since 16 is a power of two (specifically, 2 raised to the power of 4), it works well for showing these special math operations.