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Mathematical Go: Chilling Gets the Last Point

Elwyn Berlekamp and David Wolfe

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Berlekamp and Wolfe presented a new paradigm, based on Conway's theory of partisan games, for evaluating late-stage Go endgames. Expert Go players find some of the results (including the value depicted here) to be surprising and counterintuitive.

The book Mathematical Go shows how the Winning Ways style of mathematics provides new insights into a class of Go problems long known as "tedomori."

Since 1994, Berlekamp, working with Spight and other collaborators, has pursued a more extensive, and now nearly complete, mathematization of Go. This theory quantifies "tenuki." An early, rudimentary version of this "economist's" theory appears in Games of No Chance.

Extensions and refinements of this theory are now directed towards provably correct analyses of actual professional Go endgames, often lasting 100 or more moves. Simplified versions of this theory are applicable to many other games, including "Amazons".


Reviewers' comments

To get the correct solutions for problems [presented in this book] is not easy, even for professional players. However, from an analytical point of view of Mathematics, these endgame problems can be tackled from an entirely different approach which is simple, clear, complete and sequential. Being able to offer a systematic fashion of analyzing the end-game of Go is the most valuable point of this book.

Jujo Jiang, foremost Go player residing in the west (9-dan: highest rank attainable in professional Go)

This book reveals the 'secrets' to turn the late stage Go endgames into tractable exact science. It is really a treat to any serious Go player, mathematican, and computer scientist.

Ken Chen, University of North Carolina at Charlotte; author of "Go Intellect" (1994 world champion Go-playing program).

This is a most impressive book. It is the first example of a mathematical method that completely outperforms brute-force search. This book is important to anyone who is concerned with practical game theory.

Ken Thompson, AT&T Bell Laboratories; co-inventor of Unix; author of "Belle" (former world computer chess champion)

It is rare that substantial mathematics is relevant to any aspect of a popular game. Berlekamp and Wolfe are to be congratulated on a remarkable achievement.

John McCarthy, Stanford University, pioneer in Artificial Intelligence

This book is a wonderful addition to any game player's or puzzle solver's library. By analysing the last point in Go, the authors, open up, in a clear and concise fashion, a whole world of mathematical techniques and concepts for analysing games. These techniques and concepts can be applied either in a rigourous fashion or more intuitively as a help in evaluating the possibilities. Either way, they help sharpen one's problem solving abilities.

Richard J. Nowakowski, Dalhousie University, coach of outstanding Canadian Math Olympiad Teams

I'm absolutely delighted to see this book! The addition theory for partizan games grew out of my pitiful attempts to understand the game of Go. Now the failure of those attempts no longer matters, because Berlekamp and Wolfe have succeeded beyond my wildest dreams, and I heartily congratulate them.

John Horton Conway, Princeton University; renowned mathematician; discoverer of Combinatorial Theory of Partizan Games

From the back cover

In this unique book, the authors, both only amateur Go players themselves, develop the mathematical techniques for solving late-stage endgame problems that can stump top-ranking professionals. As a typical game of Go approaches its conclusion, the active regions of play become independent of one another, and the overall board position may be regarded as a sum of disconnected partial board positions. Combinatorial game theory, a branch of mathematics Berlekamp helped develop, has long been concerend with such sums of games. Here, it is applied to solving Go-related problems with a bewildering choice of similar-looking moves and subtle priority relationships. The theory presented in this book assigns each active area on the board an abstract value and then shows how to compare them to select the optimum move or add them up to determine the ideal outcome. Some of the values are familiar number or fractions, but most are bizarre objects quite unlike anything in the existing Go literature. From these abstractions, the reader learns that positions seeming to have the same numerical value can be crucially different while positions that appear completely different can be mathematically identical. A Go player with an interest in mathematics or a mathematician interested in Go will not want to miss this book because it describes substantial connections between two subject which have been, until now, largely unrecognized.

Ordering information

Books can be ordered from your bookstore or from
A K Peters, Ltd
888 Worcester St.
Suite 230
Wellesley MA 02482
781-416-2888 voice
781-416-2889 fax

Software for IBM-compatible PCs, against which you can practice your solutions to all of the problems in the Appendix of the book, is also available.

A Japanese translation of the book is available from Toppan Publishing Co of Tokyo. It includes an NEC-compatible version of the software.