On Numbers and Games

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".

At the start the only set you have is the empty set (ie the set with no members). The number 0 is defined as the result of comparing the empty set with the empty set (written as {|} ). Now you can compare the set comprising 0 with the empty set, {0|}, which is results in the number 1; you can compare the empty set with 0, {|0}, which is results in -1; and you can compare 0 with 0, {0|0}, which is the first "game" that is not a number and is called * (star).

Star is not a "game" as such but is the value for a common position in many games. In a game played between Left and Right, if the position has the value 1 (a number) then it is a win for Right, if -1 a win for Left, if 0 a win for the second person to play, and if * a win for the first player.

The book is in two, {0,1|}, parts. The zeroth part is about numbers, the first part about games – both the values of games and also some real games that can be played such as Nim, Hackenbush, Col and Snort amongst the many described.

For a more detailed treatment of the mathematics involved, see surreal numbers.

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