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Adventures in group theory. Rubik’s cube, Merlin’s machine, and other mathematical toys. 2nd updated and revised ed. (English) Zbl 1221.00013
Baltimore, MD: Johns Hopkins University Press (ISBN 978-0-8018-9013-0/pbk; 978-0-8018-9012-3/hbk). xviii, 310 p. $61.00/hbk;$ 24.50/pbk (2008).

The author states that the “book grew out of a combined fascination with games and mathematics, from a desire to marry ‘play’ with ‘work’ in a sense. It pursues playing with mathematics and working with games. In particular, abstract algebra is developed and used to study certain toys and games from a mathematical modeling perspective.” Further, the author says: “This book began as some lecture notes designed to teach discrete mathematics and group theory to students who, though certainly capable of learning the material, had more immediate pressures in their lives than the long-term discipline required to struggle with the abstract concepts involved.” The author pursues this goal by introducing a small amount of elementary logic and set theory, functions, matrices, some counting arguments, the binomial theorem, permutations and their basic properties and operations with them, the permutation puzzle. Then some puzzles are introduced, such as the hockeypuck puzzle, rainbow masterball, pyraminx, Rubik’s cube, skewb, megaminx, and a number of others. He then goes on to build up the tools that help him bridge the gap between the the play and work, namely group theory (along with a number of specific groups), group properties and constructions with groups. This bridge then takes more and more foot traffic by looking at the puzzles from the mathematical point of view, through Merlin’s machine, the orbix, lights out and its variants, finite-state machines, Gauss elimination, graphs, God’s algorithm, homotopy groups, platonic solids and 3D symmetries, the illegal cube group, group homomorphism and actions, automorphisms, the slice group of the cube and of megaminx, (semi) direct products, wreath products, free groups, words and the word problem, generators and relations, and a presentation problem. In the final chapters, the author gives a more thorough examination of the mathematics behind Rubik’s cube as well as concrete manipulations of the puzzles involved, and further tools are introduced, such as finite fields and their constructions and properties, PGL$\left(2,{F}_{5}\right)$, PGL$\left(2,{F}_{7}\right)$, Möbius transformations, the cross groups, Klein’s 4-group, the homotopy groups, the masterball group, the Mongean shuffle, PSL${}_{2}$, Mathieu groups, error correcting codes, Golay codes, quadratic residue codes, Hadamard matrices, the Hadamard conjecture, etc. The author gives mathematical solution strategies to Rubik’s cube, at the end, but these are not of performance show quality, rather they are of mathematician’s performance quality. The last chapter lists some open questions and other directions, in addition to a number of open questions and stray thoughts scattered throughout the text. An added value of this book is that it uses the computer algebra package SAGE (free to download) to do computations with groups, matrices, and everything else. Incidentally, the proceeds of the book go equally to SAGE and to the Earth Island Institute. The bibliography contains 98 references, 34 of which are web sites. There is a rudimentary index. The author peppered the text with biographical snippets and well-known mild mathematical jokes that help break the monotony of listing numerous facts and results. The book lacks a serious editorial work, but an incomplete list of glitches previously found are to be found at the following web page maintained by the author: http://www.permutationpuzzles.org/rubik/cubebook_errata2.html.

While this book is not suitable for a thorough study of any of the subjects mentioned with various levels of detail, it will give a reader an idea as to the number of mathematical topics related to some more or less well-known puzzles; an inquisitive reader then can proceed to deepen his expertise in any of the numerous subjects mentioned in the text.

MSC:
 00A08 Mathematical recreation 00-01 Textbooks (general mathematics) 20-01 Textbooks (group theory) 20B99 Permutation groups
Magma; GAP; Sage