Mathematical Treks: From Surreal Numbers to Magic Circles

This is a delightful collection of 33 items, much in the tradition of Martin Gardner, to whom it is dedicated.

How do you bridge the abyss between practising mathematicians and the general public? Of course, there is no abyss, not even a dividing line, but there is certainly a problem. Martin Gardner has been the best solution we have had for many years, and his act seems impossible to follow, but there are a few who are getting close, notable among them being Ivars Peterson.

In order to capture the reader, the trick seems to be to keep the mathematics invisible. Ivars is fairly good at this, but the technicalities are a shade more visible than they are in Gardner's writings.

Some of the perennials are there (but usually with a new twist): the Möbius strip, prime numbers, π, perfect numbers, magic squares, river crossings. Semi-popular items are Reuleaux curves, soap films, the Conway-Paterson game of Sprouts, packing circles, and Erdös. History is represented by the 1478 Treviso Arithmetic, the 1503 Margarita Philosophica and Euclid's 14th book (to be taken cum grano salis). Sporting items are the baseball diamond, the expansion draft, and ball control. Other chapters address Deep Blue, poker, dreidel, DNA, GIMPS, loci with foci, Waring's problem, spreading rumors, and matchstick problems (with a review of one of my favorite books).

All are a good read: I select four favorites, not listed above.

"Next in Line" describes the Moessner algorithm which turns additive sequences into multiplicative ones. There's a picture of the jacket of another of my favorite books.

"Mating Games and Lizards" describes how three varieties of lizard, and three strains of e. coli, play the game of Scissors-Paper-Stone.

"The Cow in the Classroom" is about humor in mathematics, citing Louis Sachar, Jon Scieszkar & Lane Smith, Stephen Leacock and many others, including Mark Twain, who calculated that a million years ago the Mississippi was 1.3 million miles long and said, "There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact."

"Computing with the EDSAC" concerns "the first fully operational and productive stored-program computer", which hasn't received the publicity given to ENIAC and other early computers. The reviewer was a user, more than fifty years ago, vicariously via the programming of C. B. Haselgrove. EDSAC calculated nim-values of impartial games. Nim-addition is just XOR, so that it's one of the fastest possible computer operations. But in those days much of the program had to be devoted to removal of the built-in "carry" that was needed for normal arithmetic! Memory size restricted calculations to 400 values, while 600 or more could fairly easily be found by hand. But the computer took a shorter time to make less mistakes, and was a valuable check. EDSAC II was still working in 1960 when my son first visited the Cambridge Lab, where he's been ever since.

There's a useful, albeit not very complete, index: a feature not always found in Martin Gardner's books. There are cleverly apt and very humorous illustrations by John Johnson. Thanks to Beverly Ruedi there are very few misprints: Gardner for Gardiner on p.67; 10n should be 10ⁿ on p.122; and on p.124, de la Vallée-Poussin's given names should be Charles-Joseph, rather than Charles-Jean.

Richard K. Guy (rkg@cpsc.ucalgary.ca) is Faculty and Emeritus Professor of Mathematics at The University of Calgary.