Combinatorial Game Theory - Examples

Examples

The introductory text Winning Ways introduced a large number of games, but the following were used as motivating examples for the introductory theory:

• Blue-Red Hackenbush - At the finite level, this partisan combinatorial game allows constructions of games whose values are dyadic rational numbers. At the infinite level, it allows one to construct all real values, as well as many infinite ones which fall within the class of surreal numbers.
• Blue-Red-Green Hackenbush - Allows for additional game values that are not numbers in the traditional sense, for example, star.
• Toads and Frogs - Allows various game values. Unlike most other games, a position is easily represented by a short string of characters.
• Domineering - Various interesting games, such as hot games, appear as positions in Domineering, because there is sometimes an incentive to move, and sometimes not. This allows discussion of a game's temperature.
• Nim - An impartial game. This allows for the construction of the nimbers. (It can also be seen as a green-only special case of Blue-Red-Green Hackenbush.)

The classic game Go was influential on the early combinatorial game theory, and Berlekamp and Wolfe subsequently developed an endgame and temperature theory for it (see references). Armed with this they were able to construct plausible Go endgame positions from which they could give expert Go players a choice of sides and then defeat them either way.