Mathematical Card Tricks
Consider a trick where the victim decides on a secret number, while your back is turned, counts off that many cards face down and looks at the last one, then replaces the dealt off cards on the top of the deck. You take the deck back, ask what the secret number was, count off that many, turning over the last card to reveal that it is not, the chosen one. ``Sorry,'' you say, again placing the dealt cards of top, ``I forgot to say `Abracadabra'.'' You deal off the secret number one final time, to find the chosen card is the last one. Not too impressive, eh? It's the old double transposition trick, which is about as profound as the fact that -(-s) = s, for all s -- known or not . But it will fool a naive audience, and suitably dressed up, the main idea forms the basis of many card tricks. The following location and prediction trick is a superb example, which never fails to impress. Forgive the old-fashioned title, we're still working on building that bridge!
Effect: The victim hides an unknown number of cards taken from a full deck, and uses that number to determine (but not remove) one card from the remainder. Taking the deck back, you claim that you'll be able to feel the selected card. However, after running more than half of the deck face down from hand to hand, you admit defeat and resort to mind reading. To the amazement of all present, yourself included, it works! This trick may be repeated, but only for an audience with a lot of patience...
Method: The performance is conveniently broken into four stages:
Typically, if you can distract the audience at the right moment, nobody has any idea that you ever peeked at a card. In fact, later on, as people try to analyze the trick, they usually swear that you could not have seen a single card (resist the temptation to shatter their illusions!). Also, few people realize that it is possible to locate their card without knowing the secret number (BTW, you still, don't know it!). Ask for the hidden cards back before you forget, and count them when nobody is looking in case somebody later says accusingly ``But you never told us how many cards were hidden!''
Mathematics: Suppose that n cards are hidden at the outset. Then the chosen card is at position n in the packet of 20 counted off the deck of 52 - n cards. The remaining 52 - n - 20 = 32 - n cards are then split into two packets, one of size m (in response to the audience's suggestion), the other necessarily of size 32 - n - m. The packet of 20 is placed on top of the packet of m, and the remaining 32 - n - m are dropped on top of these. This puts the chosen card at position (32 - n - m) + n = 32 - m from the top, and since you know m, all is well! Noting that 52 - (20 + m) = 32 - m, you can simply start counting at (20 + m) + 1 until you reach 52 to find the chosen card.
Source: This is adapted from a trick of the same name in a delightful but long out-of-print book we stumbled upon at a used bookstore in Providence, RI, during MAA Mathfest '99. It's Card Tricks Anyone Can Do (Castle Books, 1968) -- subtitled ``A Mathematical Approach to Card Magic'' on the cover page -- by Temple Patton. In the version in that book, the performer never touches the cards at all. The basic principle is of course older, as Steve Beam observes: "It is very closely related to Ed Marlo's Automatic Placement from Issue #329 of The New Phoenix published in 1955. However, Norm Houghton has also been credited with the placement - but I'm not sure how far back that goes."
Bonus points: You can start the trick by asking for somebody to call out a number between 15 and 25, and work with that many cards later, instead of 20, adjusting the counting to reflect the chosen number. This gives the illusion of less control on your part. Anyway you cut it, you are going to count out an entire deck when all is said and done, so try to divide that task up into more or less equal installments to made the process less painful for the audience. Also, other endings suggest themselves, in place of the bogus ``feeling'' or ``mind reading.'' You could dream up a long phrase to spell out to get to the chosen card, if you can do that kind of thing in your head in a hurry, or bring the card to the top of the deck and then keep it there through a few riffle shuffles, before finally producing it from behind your back (or behind somebody's ear if you can palm a card!).