theory-games

On Numbers and Games

On Numbers and Games is a mathematics book by John Horton Conway. The book is a serious mathematics book, written by a preeminent mathematician, and is directed at other mathematicians. The material is, however, developed in a most playful and unpretentious manner and many chapters are accessible to non-mathematicians.

The book is roughly divided into two parts: the first half (or Zeroth Part), on numbers, the second half (or First Part), on games. In the first part, Conway provides an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals, using a notation that is essentially an almost trite (but critically important) variation of the Dedekind cut. As such, the construction is rooted in axiomatic set theory, and is closely related to the Zermelo-Frankel axioms. Conway's use of the section is developed in greater detail in the Wikipedia article on surreal numbers.

Conway then notes that, in this notation, the numbers in fact belong to a larger class, the class of all two-player games. The axioms for greater than and less than are seen to be a natural ordering on games, corresponding to which of the two players may win. The remainder of the book is devoted to exploring a number of different (non-traditional, mathematically inspired) two-player games, such as nim, hackenbush, and the map-coloring games col and snort. The development includes their scoring, a review of Sprague–Grundy theory, and the inter-relationships to numbers, including their relationship to infinitesimals.

The book was first published by Academic Press Inc in 1976, ISBN 0-12-186350-6, and re-released by AK Peters in 2000 (ISBN 1-56881-127-6).

Synopsis

A game in the sense of Conway is a position in a contest between two players, Left and Right. Each player has a set of games called options to choose from in turn. Games are written {L|R} where L is the set of Left's options and R is the set of Right's options. At the start there are no games at all, so the empty set (i.e., the set with no members) is the only set of options we can provide to the players. This defines the game 0. We consider a player who must play a turn but has no options to have lost the game. Given this game 0 there are now two possible sets of options, the empty set and the set whose only element is zero. The game {0

is called 1, and the game {|0} is called -1. The game {0|0} is called * (star), and is the first game we find that is not a number.

All numbers are positive, negative, or zero, and we say that a game is positive if Left will win, negative if Right will win, or zero if the second player will win. Games that are not numbers have a fourth possibility: they may be fuzzy, meaning that the first player will win. * is a fuzzy game.

A more extensive introduction to On Numbers and Games is available online.