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Mathematical card tricks

There is lots of magic on the web, but not much on mathematical card tricks. The Card Tricks page at the Cyberspace Middle School website has some nice and elementary tricks, and Michael Ecker's Mathemagical Black Holes page has one nice one. Martin Gardner's 1956 classic Mathematics, Magic and Mystery (Dover) was the first book targeted at a mathematical audience which gathered some of the great mathematics-based card (and other magic) tricks in one place. Bill Simon's Mathematical Magic (Dover) from 1964 was second. Starting in the late 1950's, and continuing without a break until the early 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 this year!). Gardner's Mental Magic (Sterling) from last year, and his recent Mathematics Magazine article (May, 2000) on card tricks, 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 good specialty magic dealers -- 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. Special thanks go to Magic Castle librarian and trick inventor Gordon Bean, and author and trick inventor Steve Beam for input on the origins of some of the tricks. Thanks also to Ron Gould of Emory University, and especially to Paul Zorn of St. Olaf's College, who unwittingly got us started on all of this at a recent Joint Winter Meeting. Thanks also to Jen Chang and the Center for Experimental and Constructive Mathematics for permission to use their card-face images.

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.

``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.

Tricky business

A good card magic demonstration consists of two, or perhaps three, well-chosen and diverse tricks. Be sure to include some audience participation, never do two tricks that are likely to resemble each other from the spectators' point of view, and save your best trick until last. Leave the crowd wanting more, but resist the temptation to show them another three or four you just happen to be keen on that day: for most people a total of five or six tricks is definitely overkill. Maybe do one encore, another stunner you know never fails to impress, then quit while you're ahead -- if the audience leaves the room before you do, you know you've overstayed your welcome!

Important:

1. In general, do not tell your victim or audience in advance what (you hope) is going to happen.
2. Don't underestimate people's ability to get simple directions wrong, and save tricks which require non-trivial participation for audiences whose cerebral abilities you can personally vouch for.
3. Be crystal clear when speaking, and demonstrate anything (e.g., cutting the deck) that you expect others to do. Sometimes this also gives you a chance to pull a fast one such as peeking at a certain card, so it can serve two purposes!
4. Always have chosen cards shown around so that there can be no disagreement later as to their identity (yes, on occasion, victims have been known to forget their own special card---not to mention lie).

And always:

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. As soon as you explain it, people cease to be entertained and are inclined to think (and say!), ``Oh, is that all it is?''. The (magic) spell is permanently broken, and your stature as an entertainer is diminished in their eyes. If you are performing for intellectually curious spectators, such as colleagues or motivated students, it's different, but you still need to be careful. We strongly believe that people should, in some sense, earn the right to this kind of insider information---merely seeing you do a trick or two, being impressed, and saying ``How did you do that?'' in an endearing fashion just isn't enough!

On the other hand, being mathematically curious is a good start! 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!

Listen to a pro on the subject: "The hardest thing is to convince people that the value of a secret is in keeping it secret ... The secret of a card trick (mathematical or not) is like the punchline of a joke. Both are secrets and both are valuable in front of the current audience until you tell them. Nobody wants to hear the same punchline twice.'' (Steve Beam)

Have fun!

--Colm Mulcahy
Spelman College