On Numbers and GamesONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally written to define the relation between the theories of transfinite numbers and mathematical games, the resulting work is a mathematically sophisticated but eminently enjoyable guide to game theory. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers, that includes both real numbers and ordinal numbers. These surreal numbers are applied in the author's mathematical analysis of game strategies. The additions to the Second Edition present recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games. |
Contents
The Class No is a Field | 11 |
The Real and Ordinal Numbers | 19 |
The Structure of the General Surreal Number | 25 |
Algebra and Analysis of Numbers | 35 |
Number Theory in the Land of Oz | 41 |
The Curious Field On₂ | 46 |
Appendix to Part Zero | 60 |
FIRST PART AND GAMES | 65 |
Simplifying Games | 105 |
Impartial Games and the Game of Nim | 118 |
How to Lose when you Must | 132 |
Animating Functions Welters Game and Hackenbush Unrestrained | 149 |
How to Play Several Games at Once in a Dozen Different Ways | 169 |
Ups Downs and Bynumbers | 184 |
The Long and the Short and the Small | 201 |
Epilogue | 221 |
Playing Several Games at Once | 67 |
Some Games are Already Numbers | 77 |
On Games and Numbers | 93 |
Appendix | 225 |
Index | 231 |
Common terms and phrases
aẞ assertion atomic weight born on day Chapter component games compute conjunctive compound corresponding defined definition diagram disjunctive compound disjunctive sum dyadic rationals edges Elwyn Berlekamp equal equation field finite number follows G and H game G generalised graph Grundy number Grundy's game Hackenbush heaps impartial games inductive inequalities infinite infinitesimal integer Kayles Left and Right Left option legal move misère play negative numbers Nim-addition Nim-heap Nim-sum nodes normal play number x obtain ONAG option of G ordinal numbers ordinal sum partizan games positive numbers Proof proper Class properties prove real numbers replace reversible moves rule second player win short game sign-expansion Simon Norton simpler simplest form simplest number small games square suppose Surreal Numbers suspense number THEOREM theory thermograph trominoes Welter function winning strategy zero