Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks by Persi Diaconis and Ronald Lewis Graham

Reviewed by Jamy Ian Swiss (originally published in Genii September, 2011)

Any serious cardician is probably aware of the name Persi Diaconis, the MacArthur Award-winning mathematician, who as an adolescent travelled with Dai Vernon for a time, and would eventually write the introduction to the first edition of Vernon's legendary Revelations (1984, Magical Publications). Dr. Diaconis has in the course of his mathematics career focused on the nature of randomness with regards to subjects often of interest to magicians, such as coin flipping, and the question of how many riffle shuffles are required to randomize (depending on one's definition of randomness) a pack of playing cards. Along with Fred Mosteller (another famous mathematician and amateur magician), the two devised the Law of Truly Large Numbers, a theory which attempts to explain the occurrence of what may appear to be extremely unlikely events, such as the same person winning a state lottery more than once.

The name of Ron Graham is equally renowned in the world of mathematics, as well as in the world of not magic—but juggling. Dr. Graham served as Chief Scientist at Bell Labs during its heyday as one of America's leading technology and innovation centers; in his other life he has also served as president of the International Jugglers Association, and is also an accomplished trampolinist as part of the act known as the Bouncing Bears; and he has straddled both worlds by pioneering research in the mathematics of juggling.

These two eclectic intellectuals and variety artists have now brought all of their respective worlds together (well, except for the trampoline thing) in their new collaborative book, Magical Mathematics. Writing for the public, the two authors share their passions, teaching sophisticated mathematical concepts along with interesting card tricks which rely upon those principles for their workings. Although one chapter is devoted to juggling and obviously written primarily by Dr. Graham, the books contains much more magic and is filled with magic history and some delightful personal anecdotes, much of which obviously comes from the expertise and experience of Dr. Diaconis (who even drifts into some "inside baseball" material when discussing, for example, the publication without permission that sometimes occurs in the magic world but perhaps never in the world of academia).

While providing rather thorough introductions to mathematical principles such as de Brujin sequences and universal cycles, some of the most interesting segments draw unobvious connections between, for example, the Gilbreath Principle and Mandelbrot sets, and another chapter devotes itself to probability and the I Ching. Throughout, the authors explain many mathematically based card tricks (including contributions from Ron Wohl and Steve Freeman, some never before published) along with the theorems and proofs that provide their methodological bedrock.

As the authors correctly observe in their opening sentence, "Most mathematical tricks make for poor magic ...." Although they continue to make frequent reference to good tricks and bad tricks, they never get around much to defining the difference, which may be apparent to many magicians but will probably remain somewhat mysterious to non-magician readers. While the book provides many tricks that cleverly exploit the mathematical principles explored in tandem, what comprises a "good" trick here is a relative term. The authors have frequently performed many of these tricks for audiences of mathematicians and other academics, and for such groups who might then receive an hour's discourse on the relevant math these routines certainly do comprise "good" tricks. (And if you ever have the opportunity to perform for such groups, you want this book.) But for magicians performing for non-mathematically inclined laymen, one is compelled to ask: Do the tricks and presentations here match the bulk of material found in, for example, Steve Beam's popular books about "semi-automatic card tricks?" Probably not largely because Mr. Beam is teaching good magic that relies on math, and Drs. Diaconis and Graham are teaching math through the use of cards tricks (much as Leonardo Fibonacci did in the 13th century).

This is not necessarily a failing, because the mathematically inclined who study this book will likely be fascinated by the tricks and how the relevant math pertains to them. The question is simply one of knowing your audience. Many years ago, as a Magic Bartender in Washington, DC, I once had a handful of men come in whom I learned were attending a national conference of statisticians. One of them mentioned that he once saw a professor, whose name of course would mean nothing to me (or so he assured us), execute a shuffle which only three men in the world could duplicate. At this point, after I mentioned Professor Diaconis and a number of his contributions to the mathematical literature (which might have been more amazing to the little man in front of me than any magic I could possibly do for him), I then separated the deck into reds and blacks, and quickly executed eight out faros, restoring the deck to its original order. It is the one and only time in my life when such a feat could actually be considered unarguably entertaining what in those unique circumstances would have to be dubbed "a good trick." As the authors themselves write, some of their work "seems to play for the right kind of audience."

While I for one possess little appetite for math theory, I particularly enjoyed aspects of the book above and beyond the math. Throughout, careful credit is given to the magical creators, consistently accompanied by brief biographical details which serve to drive home the point that magicians can in fact often be a smart, eclectic, and interesting bunch. And the final chapter provides even more details about those dubbed "Stars of Mathematical Magic" (with an obvious implicit nod to our own famous Stars of Magic), including Alex Elmsley, Bob Neale, Henry Christ, Stewart James, Charles T. Jordan, Bob Hummer, and of course, last but far from least, the late Martin Gardner (who also contributes a foreword). Other entries of particular interest to magicians would include an unpublished handling of the "Endless Chain" by Stewart Judah that has long been held underground; and occasional tidbits from Dr. Diaconis's extensive collection (including hand-writ-ten notes from Ed Mario, and correspondence from Charles T. Jordan).

Some may be curious about what degree of exposure of magical methods occurs in the book, especially in light of Dr. Diaconis's reputation in this department as an infamous "black hole" repository, where magic secrets enter and from which they never escape. In these pages the good doctor is reasonably circumspect about such matters (on the one hand referring to "perfect shuffles" but never mentioning the term "Faro"; on the other, explaining the Eight Kings stack as well as the Si Stebbins system), and, remarkably, concludes the book with a positively sanguine essay about secrecy in the parallel universes of mathematics and magic. Mysteries abound!

Magical Mathematics • Persi Diaconis and Ron Graham • 6" x 10" hardcover • 2011 • 264 pages • illustrated with black-and-white photographs and diagrams • Princeton University Press