Some books make mathematics look like so much fun! This collection of 35 articles and a comprehensive bibliography is a marvelous and alluring account of a 1994 MSRI two week workshop on combinatorial game theory. This could be a menace to the rest of mathematics; those folks seem to be having such a good time playing games that the rest of us might abandon "serious" mathematics and join the party.
Even the technical terms are laced with humor. A sampling of the vocabulary of the subject includes: all small, remote star, fuzzy, domination, hackenbush, switches and tinies, hot and cold games, and loopy games. They still have their additive homomorphisms and perfect matchings, multilinear algebra, lemmas and propositions, but the fun is obvious.
The first of the books five sections is an introduction to and review of the subject of combinatorial game theory. Most readers would not be able to finish the book with only the background provided in these hundred pages, but they do point to the literature so a beginner could get a fast start, particularly to Berlekamp, Conway and Guy's "Winning Ways for your Mathematical Plays," often called just WW, and to Conway's "On Numbers and Games," ONAG for short.
The introductory section includes Julian West's account of the two tournaments featured at the workshop, one in Dots-and-Boxes, game many of us remember from childhood as "Dots," and the other in Domineering, a game partially analyzed in WW and in ONAG. How did West's dean react when he learned that West goes to meetings to play games?
The second section turns the tools of combinatorial game theory to classical games, Chess, Checkers, Nine Men's Morris and Go. Ralph Gasser describes the analysis of Nine Men's Morris that showed the game to be a draw, if both players play perfectly. Johathan Schaeffer describes analyses of Checkers and gives a great account of the best Checkers player in history, the late Marion Tinsley, and how he enthusiastically cooperated in the analysis.
The third section deals with games related to the ones in WW and in ONAG, Toads and Frogs, Pentominoes, and others. The notations get complicated and the names of the theorems get colorful, "The Death Leap Principle," "The Terminal Toads Theorem," and "The Finished Frogs Formula," for example.
Both the second and third sections rely on teamwork between the classical mathematical methodology of proving theorems and the modern use of case by case computer analyses, all guided by the experience gained by actually playing the games. Many of the results would be impossible without computers. The analysis of Checkers, for example, required assembling a database of all 443,748,401,247 positions with eight or fewer pieces on the board. On the other hand, the database would be useless without the theorems that govern the use and efficacy of proof trees. People who want to talk about the role of computers in research mathematics ought to take a close look at the interplay between human and machine that produced these results.
The fourth section describes some extensions to combinatorial game theory, games with slightly imperfect information, error correcting codes, and economics.
The fifth and last section lists 52 unsolved problems and a bibliography of 666 works in the subject.
Anyone who enjoyed either "Winning Ways" or "On Numbers and Games" will probably enjoy "Games of No Chance" as well. The writing and editing is excellent, the mathematics is interesting, and there are enough interesting games to fill thousands of napkins and placemats.
Ed Sandifer (firstname.lastname@example.org) is a professor of mathematics at Western Connecticut State University and Contributed Papers Coordinator for the North East Section of the MAA