The first book on Combinatorial Game Theory (CGT), On Numbers and Games, by John Conway, was published in the 1970s and has recently been republished. CGT was in part inspired by Go, since many Go end positions (yose) are combinations of independent regions of play.
Therefore CGT comes in naturally from the consideration of play on sub-boards. Each such position is a combinatorial game, which can be added to or subtracted from other such games. This is the idea that Go can be played on several boards at once, with alternating play still in operation: each player is able to play just one stone per turn, on one or other board
Numbers are special cases of combinatorial games: Go players can take that as meaning that some positions are already secure territory.
By convention, the players are Left (Black) and Right (White). Left's scores are positive, Right's scores are negative; we are playing a zero-sum game with a convention that plus scores are good for Left. Either player, however, may be the one to start in a game, and the notation reflects that (no convention that Black starts, therefore).
A game may be represented in slash notation (see tree representation for more about this):
{A, B, C, ... | D, E, F, ...}
To the left of the central vertical bar are the games to which Left (Black) can move, called options or followers. To the right are those to which Right (White) can move.
For example
{1 | -1}
represents a one-point gote play. (Remember that scores are games because numbers are games.) Sometimes it is written +/- 1.
In CGT the number 1 means that Left can make a play, but Right cannot (or the equivalent). Go scoring does not obviously translate to CGT numbers, but you can do it. See Mathematical Go by Wolfe and Berlekamp. Interestingly, the form of Go scoring that most straightforwardly translates to CGT numbers is territory scoring with a group tax. See Ancient Chinese Rules and Philosophy.
Anyway, 1 + (-1) = 0: the introduction of addition and equality of games is compatible with the old meanings of '+' and '='. Black can make a play, and then White can make a play, or vice versa. One way to write that game is
{ -1 | 1 }.
Most of the material originally here has been redistributed:
Original page by Bill Spight; subsequent edits by Charles Matthews.
See also: