Mathematical Card Tricks
There are four basic card handling skills needed to perform the tricks considered here: peeking, cutting, overhand shuffling, and riffle shuffling.
Peeking is exactly what it sounds like it is: the deck, which is usually kept face-down, has a bottom card whose identity can be discreetly determined at various stages in the proceedings.
Cutting the cards refers to taking any chunk of cards off the top of the deck and 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 are then 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 destroying the internal structure of the ordering. Only the top card (the start of the cycle) is altered, and if such moves are executed in the hands (rather than on 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 start out, back on the bottom -- the cards will oblige you in this manner more often than you have any right to expect!)
Overhand shuffling here means cutting over and over while the cards remain in the hands: hold the deck more or less vertically in the left hand, say, with the back of the top card facing to your right, and repeatedly use the other hand to 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 and then dovetailing together the resulting two piles, using the thumbs to release the cards, not necessarily with any great skill or regularity. By being careful which piles your thumbs let fall first and last, it is easy to maintain the top and bottom (few) cards of a deck in place; a simple observation overlooked by most spectators.
(A more specialized type of shuffle, which takes some time to master, but has
fascinating mathematical properties, is the faro shuffle: cards from
two equal packets fall alternately with total precision to give one perfectly
interwoven packet. For more information on this shuffle, and some related
tricks, we refer ithe interested reader to Solomon Golomb's ``Permutations by
Cutting and Shuffling'' (SIAM Review, Oct 1961) and the book Magic
Tricks, Card Shuffling and Dynamic Memories (Mathematical Association of
America, 1998) by S. Brent Morris. Ivars Peterson also wrote about them in
``Magic of Perfect
Shuffles'' in his August, 1998 Mathtrek column in MAA Online.)