zero game

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John Conway,Elwyn Berlekamp and Richard Guy together invented a theory for general two-player games. The term "general" here means that the game may not be identical to the two players, meaning the two players can have different game options.

Let we say the two players be L and R. A game is defined as an ordered pair (G_L | G_R), where G_L (resp. G_R) is the set of all possible configurations if L (resp. R) is the one who make a move.

A zero game 0=( | ) is the game that the first player automatically loses, because neither L nor R has a valid move.

( | 0) is the game that L must lose. If L is the first player, he has no valid move. If R is the first player, the configuration will become 0, and L being the next player has no valid move.

(0 | ) is the game that R must lose.

The star game *=(0 | 0) is the game that the second player will lose.

We have 0 game. We also have other "number" games:
1=(0 | )
2=(1 | )
3=(2 | )
4=(3 | )
....

-1=( | 0)
-2=( | 1)
-3=( | 2)
-4=( | 3)
....


It is possible to include all the numbers, some represents games with infinitely many configurations.

Invented by John Conway, the concept of games as numbers where first introduced to the public through a story book by Donald Knuth: Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. It is a rare case that a new concept is introduced this way. The name Surreal Number is given by Donald Knuth and is adopted by John Conway.

Like other number systems, the surreal numbers can add, subtract, multiply and divide.

A little exercise: For games 1,2,3,..., -1,-2,-3,...., which player (L, R, first player, second player) is the winner?

(Please also remember the name Donald Knuth, he is the creator of TeX, the standard typesetting system for most mathematical publications.)