combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the playe ...

, star, written as

*

*1

, is the value given to the game where both players have only the option of moving to the

zero game In combinatorial game theory, the zero game is the game where neither player has any legal options. Therefore, under the normal play convention, the first player automatically loses, and it is a second-player win. The zero game has a Sprague� ...

. Star may also be denoted as the

surreal form In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals ...

. This game is an unconditional first-player win. Star, as defined by

John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

in '' Winning Ways for your Mathematical Plays'', is a value, but not a

number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ...

in the traditional sense. Star is not zero, but neither

positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...

nor negative, and is therefore said to be ''fuzzy'' and ''confused with'' (a fourth alternative that means neither "less than", "equal to", nor "greater than") 0. It is less than all positive

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

s, and greater than all negative rationals. Games other than may have value *. For example, the game

*2 + *3

, where the values are nimbers, has value * despite each player having more options than simply moving to 0.

Why * ≠ 0

A combinatorial game has a positive and negative player; which player moves first is left ambiguous. The combinatorial game 0, or , leaves no options and is a second-player win. Likewise, a combinatorial game is won (assuming optimal play) by the second player

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

its value is 0. Therefore, a game of value *, which is a first-player win, is neither positive nor negative. However, * is not the only possible value for a first-player win game (see

nimber 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 '' ...

s). Star does have the property that the

sum Sum most commonly means the total of two or more numbers added together; see addition. Sum can also refer to: Mathematics * Sum (category theory), the generic concept of summation in mathematics * Sum, the result of summation, the additio ...

* + *, has value 0, because the first-player's only move is to the game *, which the second-player will win.

Example of a value-* game

Nim Nim is a mathematical two player game. Nim or NIM may also refer to: * Nim (programming language) * Nim Chimpsky, a signing chimpanzee Acronyms * Network Installation Manager, an IBM framework * Nuclear Instrumentation Module * Negative index met ...

, with one pile and one piece, has value *. The first player will remove the piece, and the second player will lose. A single-pile Nim game with one pile of ''n'' pieces (also a first-player win) is defined to have value ''*n''. The numbers ''*z'' for

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s ''z'' form an infinite

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

of characteristic 2, when addition is defined in the context of combinatorial games and multiplication is given a more complex definition.

References

* Conway, J. H., ''

On Numbers and Games ''On Numbers and Games'' is a mathematics book by John Horton Conway first published in 1976. The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpr ...

,''

Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes refere ...

Inc. (London) Ltd., 1976 {{DEFAULTSORT:Star (Game) Combinatorial game theory

Why * ≠ 0

Example of a value-* game

See also

References