Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks

To create illusions, Fibonacci and algorithms are as important as sleight of hand, discovers C.J. Howls

December 15, 2011

The brisk pace of mathematics here at the University of Southampton has been interrupted somewhat this week. Academics and students have been ambushed at every corner of the sixth floor by a shadowy figure begging to show them the card tricks that he has learned in the course of reviewing Magical Mathematics.

Over 12 chapters, the reader is taken on a unique and wonderful tour that fuses magical tricks with underlying mathematical explanations and personal stories, written by world-renowned experts in both fields. With its friendly, disarming style, the book is pitched perfectly at a level that will surprise both the hardened mathematical researcher and the interested general reader, without putting either of them off.

For example, starting with card tricks, the authors can then introduce gently de Bruijn cycles and their applications to codes and to DNA. Theorems and explanations of magic tricks naturally go hand in hand, with the first, on page 6, dealing with card-cutting. Norman Gilbreath's two principles for ordering sequences of cards are nicely linked to the Mandelbrot set.

In mathematics, if you want to learn a topic, you have to do calculations to understand it. The same is true for magic, and this book comes with a pack of cards so that readers can follow the clear textual and visual descriptions. A whole chapter is devoted to card shuffles. The authors describe how to place the top card anywhere in a standard pack by a sequence of shuffles determined simply by the binary representation of the desired location. Try this trick for yourself. It is a stunning piece of "magic", made all the more beautiful by the concealed mathematical algorithm.

Later chapters deal with more general tricks, including the divination of three (and more) objects, universal cycles, paper-folding tricks and fair-throws. Modulo arithmetic features large. The feasibility of juggling patterns is explored using "siteswap" analysis. This mathematical text must be one of few that combines detailed colour pictures on how to go about juggling with "hand-waving theorems" (the authors' joke). Another entertaining chapter deals with stars of magic (note: Harry Potter does not make an appearance), liberally sprinkled with anecdotes and further examples of mathematical underpinning, including Fibonacci sequences and geometric sequence betting in a card-simulated roulette trick.

Many of the book's tricks would be suitable for "maths busking" activities (www.mathsbusking.com) that explain underlying mathematical principles to the general public. The reader is also reminded that "magic" involves human nature, and logical explanation alone may not suffice. The authors recount how their probabilistic explanation of the Chinese Book of Changes, used for fortune-telling, resulted in a cultural clash with historians and users. Perhaps this was not surprising: the first English text on magic tricks, Reginald Scot's Discoverie of Witchcraft, published in 1584, offered a logical debunking of what was then a widely accepted belief in witchcraft. A comparison with a contemporary French text leads to a neat capture/recapture estimate of the total number of magic tricks in circulation in 16th-century Western Europe.

Psychologists might speculate as to whether there is something similar in the desire for perceived hidden knowledge and the sense of dramatic presentation that magicians and scientists arguably share. Of course, the official modern scientific method is supposed to eschew dramatic presentation because "scientific quality" alone should decide the acceptance of an idea. But can anyone really say that funding decisions are never swayed by the theatrical personality of a person making a proposal?

In summary, this book is a must-buy. If you are looking for one Christmas present for the mathematician (or mathematically minded magician) in your life, it has to be this one.

If the Royal Institution is puzzling over speakers for the 2012 Christmas lectures to excite a soporific TV audience about science, it should forget Brian Cox or Alice Roberts: send for Persi Diaconis and Ron Graham instead.

Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks

By Persi Diaconis and Ron Graham

Princeton University Press 258pp, £20.95 and £25.14

ISBN 9780691151649 and 9781400839384 (e-book)

Published 23 November 2011

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