mulcahy1 **Mathematical card tricks**

Martin Gardner's 1956 classic *Mathematics, Magic and Mystery* (Dover) was the first book targeted at a mathematical audience to gather in one place some of the great mathematics-based card (and other magic) tricks. Bill Simon's *Mathematical Magic* (Dover) from 1964 was second. Starting in the 1950's, and continuing without a break well into the 1980s, Gardner's enormously popular *Scientific American* column proved to be the perfect vehicle for further expositions along the same lines. Many of these columns on card tricks were given additional visibility over the past four decades in fifteen book collections (with another one on the way soon!). Gardner's recent *Mental Magic* (Sterling, 1999) book for children, and his ``Modeling Mathematics with Playing Cards'' article in the May, 2000 issue of *Mathematics Magazine*, are also well worth exploring.

Additional relevant tricks can be found in several excellent books by Karl Fulves (also published by Dover), and numerous slim volumes by Bob Longe (Sterling). Harder to find -- but certainly available from any good specialty magic dealer -- are Steve Beam's impressive *Semi-Automatic Card Tricks* books (Trapdoor Productions), which also contain routines of interest to the mathematically minded.

Much of the material contained here was originally organized in connection with a March 2000 MAA short course entitled ``An Introduction to Mathematical Card Tricks,'' given jointly with Jeffrey Ehme of Spelman College at the 79th Annual MAA Southeastern Section Meeting, UNC Charlotte, Charlotte, North Carolina.

We are very grateful to Martin Gardner for graciously allowing us to reproduce some of the tricks he has written about in the past, and to Gathering for Gardner co-organizer Tom Rodgers of Atlanta for his generosity with magical contacts. Special thanks go to Magic Castle librarian and magic inventor Gordon Bean, and author and magic inventor Steve Beam for input on the history and origins of some of the tricks. Thanks to Jen Chang and the Center for Experimental and Constructive Mathematics for permission to use their card-face images. Thanks also to Ron Gould and Pete Winkler, and especially to Paul Zorn of St. Olaf's College, who unwittingly got us started on this journey over a microbrewed beer during a recent Joint Winter Meeting.

## 1. Card Tricks and Mathematics

Card tricks have long been a mainstay of the magic community, and there are many fine effects using a standard deck of 52 cards that will interest lovers of mathematics. Popular techniques of card legerdemain involve forcing, controlling, location, prediction, reversal, transposition, spelling and so-called mind reading. Basic methods are counting (often secretly), reversing cards, prearranging some (or all) of the deck, and shuffling of various sorts. The tricks we discuss have underlying principles with real mathematical content, ranging from basic arithmetic, binary numbers and permutations to combinatorics and probability.

Mathematically-based card tricks can be used to liven up many mathematics classes, from precalculus and discrete math to abstract algebra, number theory and probability. Even better, such tricks are invaluable as a tool for convincing *non-mathematics students* that math can be fun and, moreover, forms the basis for certain ``real'' magic tricks (the sort some entertainers do for a living). Perhaps, by exposing students to a few tricks along these lines, we can instill them a healthy balance of respect (for the performance and entertainment aspects) and scepticism (every trick has a logical explanation) for things magical! This fits in well with the philosophy of any mathematics course which aim to equip students with quantitavive reasoning and quantitative literacy skills.

## Tricky business

``Mathematical card tricks, let it be admitted at once, are precisely the kind of tricks that are the most boring to most people,'' warned Martin Gardner as recently as 1983 in *Wheels, Life and Other Mathematical Amusements* (W.H. Freeman & Co). However, the routines we consider are unlikely to bore anyone with an interest in mathematics, and even professional magicians have been known to perform them accompanied by appropriate show business trappings (and without advertising the tricks' mathematical underpinnings).

The person you perform a trick for, who often assists in some way, is usually referred to as the *victim*. There are four basic card handling skills needed to perform the tricks considered here: *peeking*, *cutting*, *overhand shuffling*, and *riffle shuffling*. If these sound familiar, you are ready to proceed to the first trick. Otherwise, click on the link above.

Please see Tips of the trade for some important words of advice on magic performance and ethics. In particular, remember the

Golden Rule of Magic --- *Never reveal your secrets!* If you are performing for a ``lay audience,'' you really should think twice before divulging how a trick works. If you are performing for *intellectually curious* spectators, such as colleagues or motivated students, it's different. Give people hints and let them work out some of it themselves, it's *far* more rewarding than being handed the whole trick on a plate. *This applies to you too!* -- try to work out the tricks as you go along, you'll be glad you did!

It is also a mistake, in general, to repeat a trick for an audience, even if asked to. There are exceptions, of course, which we note.

Have fun!

-- *Colm Mulcahy *

Spelman College