**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:

- Invite somebody to shuffle the deck, and have the victim 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 peprson can hold up an appropriate
number of fingers while your back is turned). This is the
*key number*, which everybody but you knows. The removed cards 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 nth 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 52nd 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 mid-valued 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!''

**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!).

- 1. Card Tricks and Mathematics
- 2. Numerology it isn't
- 3. Subtraction is addictive
- 4. It's as easy as one, two, three
- 5. Binary 101
- 6. It's probably magic
- 7. (Smells like) Team spirit
- 8. Tips of the trade
- Appendix. Basic card handling skills

@ Copyright 2000, American Mathematical Society.