On Numbers and Games - 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 {|}, 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.

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

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