# On Numbers And Games

John Conway

## Prologue

Just over a quarter of a century ago, for seven consecutive days I sat down and typed from 8:30 am until midnight, with just an hour for lunch, and ever since have described this book as "having been written in a week."

Not entirely honest, because there were loose ends still to be tied up, and Chapter 16 was written just before the book appeared, while Chapter 13 was largely copied from a paper, "Hackenbush, Welter and Prune", that had been written a year earlier. But also not entirely dishonest.

Why the rush? Because ONAG, as the book is familiarly known, was getting in the way of writing *Winning Ways* (WW). Now that both books are happily being republished by A K Peters, Onagers (a word that also means "Wild Asses"!) can be told just how it came about before they surrender themselves to pure pleasure (as "Onag" means in Hebrew!).

A few years previously, Elwyn Berlekamp, Richard Guy and I had agreed to write a book on mathematical games, by which at that time we meant the Nimlike theory developed independently by Roland Sprague and Peter Michael Grundy for sums of impartial games-those for which the two players have exactly the same legal moves.

I had long intended to see what would become of the theory when this restriction was dropped, but only got around to doing so when the then British Go Champion became a member of the Cambridge University Pure Mathematics Department. Astonishingly, it was the resulting attempt to understand "Go" that led to the discovery of the Surreal Numbers! This happened because the typical "Go" endgame was visibly a sum of games in the sense of this book, making it clear that this notion was worthy of deep study in its own right. The Surreal Numbers then emerged as the simplest domain to which it applies!

However, their theory rapidly burgeoned in ways that made it inappropriate for the book that later became *Winning Ways*. A busy term was approaching, and it seemed that this "transfinite" material just had to be got out of the way before that term started if *Winning Ways* was ever to be published. So I sat down for that week and wrote this book, and then confessed the fact to my co-authors.

The most surprising immediate result was a threat of legal action from Elwyn Berlekamp! But somehow we must have patched this up, because both ONAG and WW appeared in the next few years, and we remain good friends.

In fact, the Surreal Numbers "surfaced" before ONAG appeared, partly through my 1970 lectures at Cambridge and Cal. Tech., but mostly through the wide circulation of Donald Knuth's little book, *Surreal Numbers*. I am very grateful to Knuth for inventing this name-the original version of ONAG said "Because of the generality of this Class, we shall simply describe its members as numbers, without adding any restricting adjective." "Surreal Numbers" is much better!

I am very happy and grateful that A.K. Peters have agreed to publish millennial editions of both this book and *Winning Ways*.

Ariel Jaffee and Kathryn Maier were responsible for handling the changes to this edition. This is also the place to acknowledge Richard Guy's considerable contributions to the original edition. In particular, he designed and drew a number of the original figures and computed, or recomputed several of the tables.

I have called this a Prologue rather than a Preface because it is usually understood that the Preface to a later edition of a book should contain a Description of the changes in the book and its subject since its first edition. Some of these functions are addressed in the Epilogue.

John H. Conway

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