On Numbers and Games

ONAG, 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.

Prologue

v

Preface

vii

Zeroth Part...On Numbers

1

(68)

All Numbers Great and Small

3

(12)

The Class No is a Field

15

(8)

The Real and Ordinal Numbers

23

(6)

The Structure of the Genral Surreal Number

29

(10)

Algebra and Analysis of Numbers

39

(6)

Number Theory in the Land of Oz

45

(5)

The Curious Field On2

50

(19)

Appendix to Part Zero

64

(5)

First Part...and Games

69

(156)

Playing Several Games at Once

71

(10)

Some Games are Already Numbers

81

(16)

On Games and Numbers

97

(12)

Simplifying Games

109

(13)

Impartial Games and the Game of Nim

122

(14)

How to Lose when you Must

136

(17)

Animating Functions, Welter's Game and Hackenbush Unrestrained

153

(20)

How to Play Several Games at Once in a Dozen Different Ways

173

(15)

Ups, Downs and Bynumbers

188

(17)

The Long and the Short and the Small

205

(20)

Epilogue

225

(4)

Appendix

229

(6)

Index

235

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Summary

Table of Contents

Supplemental Materials