October 2005
Consider a trick where the volunteer 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 reversal 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 the card tricks explored here this month.
Predictable Difference
You write a prediction on a piece of paper, and hand a shuffled deck to a volunteer, requesting that any number of cards be counted off face down into a pile on the table. The next card is turned over and checked against your prediction — it does not match! The deck is reassembled and dealt again, the volunteer counting off even more cards this time. Once again the next card is turned over, but once again it fails to match your prediction. Finally, you suggest that the volunteer spell out a magical word, while counting out cards face up. Sure enough, when the next card is turned over, your written prediction is finally vindicated.
You must know the identity of the top card at the outset; this you write down as your prediction. You can riffle shuffle the deck repeatedly without disturbing this setup. Count carefully (to yourself) as the volunteer deals out cards face down the first time, and look crestfallen when the next card fails to match your prediction.
Have the card replaced on top of the deck, followed by the dealt cards. Count once more as the volunteer deals the second time, ensuring that several more cards are counted out. Express exasperation that the next card also fails to match your prediction. Have that card, and then the dealt cards, replaced on top of the deck. You now have two numbers in your head, one for each count/deal.
Subtract the smaller number from the larger one, and quickly think of a magical word or phrase with that many letters (e.g, "LUCK," "MAGIC," "MY CARD," "MAGICAL," "IT'S MAGIC," etc.). Have the volunteer do one last deal, counting out one card for each letter. The next card will match your prediction.
Why does this work? Suppose the first deal involved s cards, and the second deal t cards, for some t > s. After the first deal and reassembly, your prediction card (the original top card) is in position s (from the top), which one can check moves to position t  s + 1 after the second deal and reassembly. As you secretly keep track of s and t, you can easily direct the spelling out of a word or phrase with t  s letters. The next card will be the desired one (i.e., the original top card).
This is slightly modified from "The Keystone Card Discovery" in Martin Gardner's Mathematics, Magic and Mystery (Dover, 1956), where it is traced back to 1920. A trick based on the same principle is "Lucky Numbers" from Bob Longe's Mystifying Card Tricks (Sterling, 1997).
Next we present the best application of the subtraction principle we've encountered, which allows for a great variety of presentations and works as well with a large crowd as a small one. It was stumbled upon in the dying days of the century it's named after. Don't let the scent of moth balls put you off.
Twentieth Century Mind Reading
A volunteer hides number of cards — unknown to you — 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 from hand to hand, face down, 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.
The performance is conveniently broken into four stages:

Invite somebody to shuffle the deck, and have a volunteer decide on an integer n between 1 and 20, discreetly remove that many cards, and pass them around so that others can secretly count them (or that person can hold up an appropriate number of fingers while your back is turned). This is the key number, which everybody but you knows. The cards removed remain hidden from you for the remainder of the performance. Now take back the rest of the deck, and show the audience the faces of the first 20 cards, one by one, while retaining their order, instructing one and all to spot and remember the n^{th} card, but not to react in any way when they see it go by. (You could say, "This is really a psychological test. It's important that you look just as bored at the beginning as you will feel by the end.")

Put those 20 cards down, and ask for a random number, perhaps suggesting, "About half of what I have left here," holding up the balance of the cards. If somebody calls out "Seven," count out 7 cards, place the packet of 20 on top of them, and drop the lot on top of the remaining cards, remarking "It's important that I don't know how many cards are left, because then I could work out how many you hid at the start, and we don't want that do we?"

Now comes the show business: start passing cards from the top of the deck, in your left hand, to your right hand, again retaining the order. Make it clear that you cannot see the faces as you pretend to "feel" them, but don't let anybody else see them either. What you really do is silently count to 52 as you pass cards, starting with 20 + 7 + 1 = 28 ("28, 29, ..., 52"). The 52^{nd} card passed is the chosen card, and it is now on the bottom of the pile in your right hand! Place the cards in your left hand on top of those in your right hand, thus bringing the chosen card to the bottom of the deck, while distracting the audience by remarking, "It's not working, I just don't have a feel for your card today. Besides, you sure don't want to let me get to the end of the deck in case I was trying to figure out your secret number!"

As you say this, square up the deck by tapping it on the table, ensuring that you and you alone get a peek at the bottom card. Cut the deck, riffle shuffle, and push it to one side. Suppose the chosen card you just spotted is 6 Clubs, proceed something like this: "Let's resort to mind reading ... the lady in the back with the glasses is giving me vibes ... a black card you say?" Even though you had a 50% chance of getting that right, the audience is likely to grow quiet and hang on your every word from now on. Direct your attention to somebody who looks like they are muttering something skeptical to a neighbor, and point, saying "Hmm, you're telling me it's a midvalued black card — and even!" Look at somebody else, "You sir, I get the distinct impression you have six on your mind. So it's a black six!" Complete the revelation quickly with the "help" of somebody else.
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!"
Why does this trick work? 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.
This is adapted from a trick of the same name in a delightful, long outofprint book called 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, stumbled upon at a used bookstore in Providence, RI (during MAA MathFest '99), the performer never touches the cards at all.
The basic principle is older, as magician 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."
You can also start the trick by asking somebody to call out a number between 15 and 25, and work with that many cards 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 make the process less painful for the audience.
Other endings also 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 on the fly, or bring the card to the top of the deck and then keep it there through a few riffle shuffles, before producing it from behind your back (or behind somebody's ear if you can palm a card!).
We wrap up with another classic from bygone days in which subtraction is the real brains behind the operation. We predict that it will continue to impress for at least another hundred years.
A Twenty First Century Force
A volunteer is handed a deck, and asked to cut off between a quarter and a third of the cards. Have these counted silently, and while this is going on, deal out a long row of cards from the rest of the deck. Turn away, and ask the volunteer to peek at the card whose position in the row matches the number of cards cut off. Have the entire deck reassembled and thoroughly shuffled. Turn back and scan the card faces. Shuffle again and ask what the chosen card was. Say, "I thought as much," as you turn over the top card to reveal that it matches the card just identified.
The card in position 21 is a force card, which you must know ahead of time. Proceed as above, and while the volunteer is counting the cutoff cards, line up exactly twenty cards on the table, reversing their order. (Deal the row from right to left.) When the volunteer peeks at the card whose position matches the number of cards cut off — making sure that the counting is done from left to right — it will be the force card. The rest is automatic: when you scan the faces later, simply cut the force card to the top and keep it there through a few riffle shuffles.
Suppose the original deck is numbered: 1, 2, 3, ..., k, k+1, ..., 20, 21, ..., 52, where k is, say, between 10 and 18. The first k cards are taken in a single clump by the volunteer (it is this action which determines k). You now deal out (and hence reverse) the top 20 cards from the 52  k left, resulting in the row numbered k+20, k+19, ..., k+1, from left to right. (Set aside the remaining (52  k)  20 cards, they are no longer needed). The t^{th} card in the row on the table is numbered (k + 20)  (t  1). In particular, the k^{th} card is (k + 20)  (k  1) = 21, the original 21^{st} card in the deck.
This principle is very wellknown, and is related to another popular magic principle, the Clock Force. It was studied and generalized by Ed Marlo and Stewart James. A nice variation is to force the 26^{th} card instead, dealing out a row of 25 cards to do this. If you are good at splitting the deck in half exactly, perhaps as a result of trying to master faro shuffling, then you can do this and hand out the two halves to two people to be shuffled freely, peeking at the bottom of the top half just before the deck is reassembled. May the force be with you.
The moral of this month's colm is that subtraction is addictive, even though we've yet to find a way to take the 20th century from the 21st century.