**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.

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.

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

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

- 1. Card Tricks and Mathematics
- 2. Numerology it isn't
- 3. Subtraction is addictive
- 4. It's as easy as one, two, three
- 5. Binary 101
- 6. It's probably magic
- 7. (Smells like) Team spirit
- 8. Tips of the trade
- Appendix. Basic card handling skills

@ Copyright 2000, American Mathematical Society.