**Mathematical Card Tricks**

Binary searches and sorts form the basic for many fine card routines, including
the next **forcing and prediction trick**, which utilizes the so-called
*down under deal*: a packet of cards is dealt out by first dealing a card
onto the table, then one underneath the cards remaining in your hand, and so
on, until only one card remains.

**Method:**
Method: You must know the top card on the deck at
the outset. This can be peeked at ahead of time, and some
distracting (but harmless) riffle shuffles executed. Next, you need to compute
twice the difference between
the number called out and the highest power of two which is strictly less
than it. E.g., if fourteen is called out, compute
2 * (14 - 8) = 12. Now deal out that many cards casually
into a pile, thus reversing their order. Gather up the
cards while feigning poor memory, saying
``How many did you say? Oh, fourteen,'' and scoop the extra two cards off
the deck and place them beneath the cards in your hard.
In this way you now have the card you know in the twelfth
position in a packet of fourteen cards.
If, on the other hand, sixteen is called out,
simply deal that number into a pile.
In either case, hand this packet to the victim,
and carefully direct the down under dealing.
The last card is guaranteed to be the original top card.

There are other, less obvious, ways to achieve the goal of the false count above which will likely go unnoticed by your audience -- just use your imagination.

**Mathematics:** Everything becomes clearer if we convert
to base 2. Suppose the called out number is 1abc...e in base 2.
An exercise for the reader is to show that the down under deal
from a packet of 1abc...e cards always leaves as
last card the card
which started in position
abc...e0, unless the number of cards in the packet is exactly a power of two;
and in that case
the down under deal ends up with the original bottom card in the packet.
Now if 1abc...e is not a power of two, the highest
power of 2 strictly less than it is 1000...0, and when this is
subtracted from 1abc...e we get abc...e. Twice
that is abc...e0. On the other hand, if 1abc...e = 1000...0 is a power of two,
the next highest
power of two is 100...0 (one
less
zero), the difference is also 100...0, and twice the difference
gets back the total number of cards in the packet.
Either
way, the down-under deal will end up with the card you peeked at.

**Source:**
Gardner mentions learning the basic idea behind this one from Mel Stover in
*The Unexpected Hanging and Other Mathematical Diversions* (Simon and
Schuster, 1969, republished by Univ. of Chicago Press, 1991), and credits
magician John Scarne with the above presentation. He also mentions a Bob Hummer
version from 1939, and describes several variations on the theme due to Sam
Schwartz and others.

**Bonus points:**
An easier trick, dating back to the 1920s, which uses the fact that each number
is a sum of powers of 2, can be found in Gardner's *Mathematics Magic and
Mystery* (Dover, 1956), listed as ``Findley's Four-Card Trick,'' and in
Simon's *Mathematical Magic* (Dover, 1964), where it appears in the guise
of ``The Spirit Mathematician.''

*Revised 11/25/00*

- 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.