Tricky Mathematics

March 18, 2012

Book Review
Philip J. Davis


Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks. By Persi Diaconis and Ron Graham, Foreword by Martin Gardner, Princeton University Press, Princeton, New Jersey, 2011, 258 pages, $29.95.

Our whole life is solving puzzles.---Erno Rubik

Some years back at a large family gathering, a third cousin twice removed, a relative whom I hadn't seen in years, recognized me and approached with a deck of cards in his hand. "Take any card," he said, "look at it, but don't tell me what it is. Then put the card back in the deck, anywhere." I complied, after which he gave the deck a few taps and then produced my card. "Wow," I said, feigning surprise, and walked away to where the drinks were.

Because of this admitted indifference, I am not your ideal reviewer for this book, which was written by two mathematicians of international reputation; masters of the arts of mathemagic, they have put on shows along those lines. My beau ideal of magic is when the magician produces a rabbit from his hat or saws a scantily clad maiden in two. Although I have enjoyed such shows, they rarely leave me curious enough to ask, "How is that done?" Ditto for mathemagic. I am willing to allow a craft its secrets, but Diaconis and Graham have been more than generous in revealing their smoke and mirrors.

Nonetheless, it is clear to me that Magical Mathematics is a treasure trove of "The Mathematical Ideas that Animate Great Magic Tricks" and hence an excellent source for people who like to perform such things. Simultaneously, via the histories and biographies embedded in the book, particularly in the chapter titled "Stars of Mathematical Magic," the book is a window into the day-to-day activities of the mathemagicians of the world, and a number of famous practitioners and their art are mentioned by name. The historical section gives a nod to J. Prevost, French author of one of the earliest serious magic books (1584), and more than a nod to his slightly later counterpart, Gaspard Bachet (1612). If you want more dates and personalities, flip the pages of David Singmaster's Chronology of Recreational Mathematics, where the timeline goes from before 2000 BC to 1996.

Recreational mathematics has long had its constituencies: those who love it, create it, promote it, publish it, historicize it, and even critique it. In August 1988, for example, the List Art Center at Brown University hosted a show of the many magic squares, magic discs, stars, polyhedra created by Royal V. Heath, a man dissed in the book under review---tedious and boring.

Along with historian of mathematics Eric Temple Bell, one might think of mathematics as "the handmaiden of the sciences," or of economics, or even of sports or politics, but the four-millennium-long history of the field displays the wide variety of the uses of mathematics, of its purposes or modes of thought. In the coinage of anthropologist Stephen Jay Gould, these are varieties of mathematical "magisteria," i.e., domains of authority, in which this grand human invention plays out.

There is Pythagorean number mysticism, e.g., two is feminine, three is masculine. There is religious or ritual numerical mysticism, e.g., the Trinity, the Number of the Beast---666---in the Book of Revelation. There is Gematria, in which words are assigned numbers and the relation between the numbers feeds back into the words. There is "septolatry," which, as set forth in numerous writings, shows how you can connect the number seven with everything significant. There is calendric numerology or divination, as practiced by John Napier (of logarithmic fame), who used intervals of dates to predict the Second Coming and the End of the World. This could well be called "mathematical eschatology," a fairly recent example of which is the entropic eschatology of Oswald Spengler, who speculated about the heat death of the universe. Last, but hardly least, there is recreational mathematics, as in Magical Mathematics. In the books of mathematical history that I have at hand, the only mention of the word "magic" I could find was in the term "magic squares." Yet such commentators as Yuri I. Manin have implied that all of mathematics is by its nature magic.

Recreational mathematics might be defined as mathematics for amusement and is often pursued via a variety of "tricks." Many such are elaborated in the book under review. Here are but a few of the things you will find: the I Ching and its magical aspects, de Bruijn sequences, patterns of juggling balls, paper folding, tiling, chain throwing, Hamiltonian cycles, roulette systems with cards, card shuffling, including "a riffle shuffle card trick and its relation to quasicrystal theory."

Though not a practitioner, I have been aware of the existence of mathemagic for a long time. When I was about ten years old, a set of old books mysteriously appeared atop the piano in our living room. Here is the full title of the set: Rational Recreations: In Which the Principles of Numbers and Natural Philosophy Are Clearly and Copiously Elucidated, by a Series of Easy, Entertaining and Interesting Experiments among which are all those commonly performed with the Cards. Author: William Hooper, MD. Date: 1774.

How these volumes got there I never found out, but I assume that my older brother, who already had a degree from MIT, had bought them. Not in the least theoretical, these volumes begin with recreational arithmetic and then cover the waterfront of experimental chemistry, physics, optics, chromatics, acoustics, pyrotechnics (i.e., fireworks), etc.

A clip from Volume 1 of Hooper (Arithmetic-Mechanics, 3rd Edition, 1787):

"Recreation XIII. Thirty soldiers having deserted, fifteen of whom are to be punished, so to place the whole number in a ring so that you may save any fifteen you please and it shall seem to be the effect of chance."

This is a variant of the old Josephus Problem.

The contrast between the work of Hooper and Diaconis–Graham is vast. In the former, the tricks involve elementary arithmetic, most of which a sixth grader should be able to see through. In the latter, the mathematics is very far from grade school and the authors have suggested open research problems.

Recreational mathematics has been around for ages and has an assured future. If you are as clever and as fortunate as Erno Rubik (of Rubik Cube fame), you may be able to devise a new trick, a game, or a device that will earn you a fortune and that the cognoscenti will be able to link to the latest mathematical theories.

Philip Davis, a professor emeritus in the Division of Applied Mathematics at Brown University, has been reviewing books for SIAM News for more than twenty years. At the same time, he has combined a lifetime of mathematics, scholarship, and abundant gifts as a raconteur in writing many books of his own.



The most recent, Ancient Loons: Stories David Pingree Told Me,* is, as its title suggests, a collection of stories recounted to Davis by his longtime Brown colleague David Pingree---a historian of ancient mathematics and long a member of Brown's now defunct Department of the History of Mathematics. With additional twists of his own, the ever curious and engaging Davis retells the stories, which, according to the publisher, "trace connections between ancient characters, historical and mythical, and recreate a world in which the pursuit of knowledge for its own sake leads to unexpected pleasures and associations."

A copy sent to
SIAM News at Christmas quickly disappeared into the briefcase of SIAM executive director (and Brown alumnus) Jim Crowley, who returned it not long afterward."This remarkable little book by Phil Davis brings together bits of philosophy, history, and mathematics," he said. "There are few people today who could pull all this together and make it entertaining."

*An AK Peters Book, published by CRC Press.


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