More Games of No Chance

School Science and Mathematics, Nov 2004 by Ratcliff, Gail

More Games of No Chance Author Richard J. Nowakowski, Editor Cambridge University Press 40 West 20th Street New York, NY 10011 2002; 535 pages Hardback $55.00

Games of no chance, or combinatorial games, are those that involve no dice or spinners or hidden cards. Both players have full information at the start of the game. As described by Nowakowski,

This book is a state-of-the-art look at combinatorial games, that is, games not involving chance or hidden information. It contains articles by some of the foremost researchers and pioneers of combinatorial game theory...The articles run the gamut from new theoretical approaches...to the very latest in some of the hottest games...Many of these advances reflect the interplay of the computer science and the mathematics.

The reader of More Games of No Chance should begin with the final article, "Combinatorial Games: Selected Bibliography With a Succinct Gourmet Introduction" by Aviezri Fraenkel, which gives a brief description of the subj ect and its relationship to popular games like Chess, Go, and applications to complexity theory. The natural appeal of games seems clear to Fraenkel: "Perhaps the urge to play games is rooted in our primal beastly instincts; the desire to corner, torture, or at least dominate our peers." This short article is followed by a bibliography of 919 items - it has been under construction for 20 years.

For non-experts, a great introduction to the subject is the two-volume classic Winning ffoysbyBerlekamp, Conway. and Guy. Another classic is Conway's book, On Numbers and Games.

Practitioners of combinatorial game theory like to give witty names to their inventions, and puns abound. Fraenkel describes "Wonderland" - the place where most games live because "we are still wondering about their as yet unknown complexity." These researchers are likable characters. John Conway notes, for instance, that "I am also interested in wasting time, professionally, and I have invented some very powerful ways of wasting time. One of these is combinatorial game theory." Refreshing words from a professor of mathematics at Princeton University!

Although most of the articles will not be accessible to teachers and their students, one can learn the rules for many games by reading the first page or two of each article. These games have names like Nim, Hex, Chomp, Amazons, Domineering, Dot-and-Boxes, Konane, and Phutball.

Combinatorial games provide the teacher with a creative means to allow students to explore mathematical ideas and develop problem-solving skills. While the rules are simple, there are rich mathematical theories underlying these games. Students are puzzled at first, and seem to make random moves. By encouraging them to start with simple games with a small number of pieces and then gradually increase the complexity, students are able to formulate and test their own theories for strategies and solutions.

Games are innately appealing to students (and researchers!) at all levels, and can be used to create lively and stimulating classes.

Editor's Note: S. WaIi Abdi's postal address is The University of Memphis, Department of Instruction and Curriculum Leadership, 401 A Ball Hall, Memphis, TN 38152, ande-mailaddressisabdi.wali@coe.memphis.edu

Reviewer

Gail Ratcliff

Department of Mathematics

East Carolina University

Greenville, NC 27858