This is the second volume in The New Martin Gardner Mathematical Library, a series that will reprint all the collections of Gardner's "Mathematical Games" columns. For a discussion of the series as a whole, see the review of the first volume.
Origami, Eleusis, and the Soma Cube is the new title for what was published, in 1961, as The 2nd Scientific American Book of Mathematical Puzzles and Diversions. This means, of course, that these columns were first published in the late 1950s. For all their age, most of them remain fresh and interesting. Gardner has provided updates as appropriate, and the bibliographies have also been brought up to date.
One of Gardner's real strengths as a writer of popular mathematics is his ability to choose topics that will appeal to the general public while also having some real mathematical content. The stand-out example in this collection is the chapter on Origami. There are real mathematical possibilities implicit in the ancient art of paper-folding, but when Gardner wrote his column they seem to have been largely unexplored. Fifty years later, there has been something of an explosion of interest and new ideas. These are briefly mentioned in Gardner's update, and the very selective bibliography for this chapter runs to four pages.
While Origami has stayed in the limelight, other topics discussed by Gardner have not. This doesn't mean they are any less interesting. Eleusis, for example, is a card game designed to require players to construct guesses based on the evidence before them. If I ever teach a course on the history of science, I may well teach the class the game to give them a sense of how inductive reasoning works.
I found the chapters on "The Monkey and the Coconuts" and on "Probability and Ambiguity" particularly fascinating. The first tells the story of a problem mentioned in a short story that caused a furor. The second deals with the difficulty of coming up with unambiguous statements of various probability problems. It includes a problem that is equivalent to the infamous "Monty Hall Problem," and the solution is given correctly.
The column on the soma cube includes two (unintentional) name-drops that I found amusing. The first is a reference to "Richard K. Guy of the University of Malaya in Singapore." Guy, now retired from the University of Calgary, is in his 90s and still going strong. The other is a reference to "Mrs. R. M. Robinson, wife of a mathematics professor at the University of California at Berkeley," who, of course, is none other than Julia Robinson. I was reminded of the delightful story in Julia, a Life in Mathematics in which the mathematics department at Berkeley gets informed that "Prof. Robinson's wife" had just been elected to the National Academy of Sciences.
The obligatory discussion of the golden ratio is here, and to his credit Gardner was already somewhat skeptical of the standard stories surrounding it. In his update, he says that he has since become even more skeptical, and rightly so. Equally interesting, though now very much out of date, is the chapter on mechanical puzzles. And who can resist reading about "Recreational Topology"?
One unusual column should be mentioned. "Squaring the Square" is not by Gardner at all, but by William T. Tutte. It tells the story of how the problem of dividing a square into finitely-many non-congruent subsquares was solved. The idea was to reinterpret it as a problem in graph theory, and then to reinterpret that as a problem about electrical circuits. It's a neat story. One of the things that struck me as I read it is that most mathematicians today would be able to do the first step (from dissection to graph theory) but would never think of the second, since most of us do not know Kirchhoff's Laws.
There's a lot more in there. Like the first volume, this one is in the "must have" category.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. When this book was first published, he didn't know how to read yet. He has since learned.