Mathematical Card Tricks
Appendix: Basic card handling skills
There are four basic card handling skills needed to perform the tricks consideredhere: peeking, cutting, overhand shuffling, and riffleshuffling.
Peeking is exactly what it sounds like it is: the deck, which is usually keptface-down, has a bottom card whose identity can be discreetly determined at variousstages in the proceedings.
Cutting the cards refers to taking any chunk of cards off the top of the deckand setting it on the table beside the rest. It is usually understood that``the cut is complete,'' which is to say that the original bottom cards arethen placed on top of the cards cut off and the packet squared up. Mathematically,it is clear that this operation merely cycles the entire deck around, in order:a prepared deck or packet may be cut like this repeatedly without destroyingthe internal structure of the ordering. Only the top card (the start of thecycle) is altered, and if such moves are executed in the hands (rather thanon the table), it is not so difficult to get the original top card back on top.(Just get the original bottom card, which you can sneak a peek at as you startout, back on the bottom -- the cards will oblige you in this manner more oftenthan you have any right to expect!)
Overhand shuffling here means cutting over and over while the cards remain inthe hands: hold the deck more or less vertically in the left hand, say, withthe back of the top card facing to your right, and repeatedly use the other handto lift off packets of cards from the top and drop them on the bottom.
Riffle shuffling refers to the act of splitting the deck roughly in half andthen dovetailing together the resulting two piles, using the thumbs to releasethe cards, not necessarily with any great skill or regularity. By beingcareful which piles your thumbs let fall first and last, it is easy to maintainthe top and bottom (few) cards of a deck in place; a simple observationoverlooked by most spectators.
(A more specialized type of shuffle, which takes some time to master, but hasfascinating mathematical properties, is the faro shuffle: cards fromtwo equal packets fall alternately with total precision to give one perfectlyinterwoven packet. For more information on this shuffle, and some relatedtricks, we refer ithe interested reader to Solomon Golomb's ``Permutations byCutting and Shuffling'' (SIAM Review, Oct 1961) and the book MagicTricks, Card Shuffling and Dynamic Memories (Mathematical Association ofAmerica, 1998) by S. Brent Morris. Ivars Peterson also wrote about them in``Magic of PerfectShuffles'' in his August, 1998 Mathtrek column in MAA Online.)