On Numbers and Games

On Numbers and Games is a mathematics book by John Horton Conway. The book is a serious mathematics book, written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a most playful and unpretentious manner and many chapters are accessible to non-mathematicians.

The book is roughly divided into two sections: the first half (or Zeroth Part), on numbers, the second half (or First Part), on games. In the first section, Conway provides an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals, using a notation that is essentially an almost trite (but critically important) variation of the Dedekind cut. As such, the construction is rooted in axiomatic set theory, and is closely related to the Zermelo–Fraenkel axioms. Conway's use of the section is developed in greater detail in the Wikipedia article on surreal numbers.

Conway then notes that, in this notation, the numbers in fact belong to a larger class, the class of all two-player games. The axioms for greater than and less than are seen to be a natural ordering on games, corresponding to which of the two players may win. The remainder of the book is devoted to exploring a number of different (non-traditional, mathematically inspired) two-player games, such as nim, hackenbush, and the map-coloring games col and snort. The development includes their scoring, a review of Sprague–Grundy theory, and the inter-relationships to numbers, including their relationship to infinitesimals.

The book was first published by Academic Press Inc in 1976, ISBN 0-12-186350-6, and re-released by AK Peters in 2000 (ISBN 1-56881-127-6).