On Numbers and Games

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On Numbers and Games is a mathematics book by John Conway, published by Academic Press Inc in 1976, ISBN 0121863506, and re-released by AK Peters in 2000 (ISBN 1568811276).

It uses sets to define a universe of numbers (whole numbers, fractional numbers and some very strange surreal numbers of no use to the general public) and as a by-product, uses the same method to define some "games".

A game in this sense 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.[1] 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 {|}, which is called 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.

The book is in two, {0,1|}, parts. The zeroth part is about numbers, both real numbers that people use and some strange surreal numbers that mathematicians dream of. The first part is about games, both the mathematical analysis of games and several real games that can be played for enjoyment. Nim, Hackenbush, Col and Snort are among the many games described.

  1. ^  Alternatively, we often list the elements of the sets of options to save on braces. This causes no confusion as long as we can tell whether a singleton option is a game or a set of games.

See also: Winning Ways for your Mathematical Plays, Combinatorial game theory.