**Mathematical Card Tricks**

Bearing in mind the unavoidable possibility that a trick which is known to be mathematically watertight can go wrong -- either the magician or a victim can make a mistake and cause the whole effect to blow up in your face -- it's as well to be prepared for emergencies and keep a short, snappy crowd pleaser in reserve that you know cannot fail. It's also worth considering tricks that merely have a high probability of working. Even if mathematics cannot guarantee 100% success every time, the payoffs when the tricks do work arguably outweigh the red faces on the rare occasions when they don't. If the dice seem to be against you on the day that you try one of these in public, simply act as you do when one of your old reliable tricks lets you down, and say nothing to make your audience think this trick is any different from the rest.

This brings us to one of the most startling **prediction** tricks imaginable.

**Method:**
The method is simple: you yourself follow the same instructions as the victim
deals off the cards, picking at random some card among the first ten, and so
on, finally arriving at your own last key card. The amazing thing is that
with very high probability you'll both have come to the same final key card.
In fact, your apparantly independently determined streams will most often
flow together somewhere along
the way, sometimes before you are even half way through the deck!

We advise you to run through a full demonstration of the counting procedure for the victim, pausing at each key card for clarity, before you let them loose with the deck (don't forget to shuffle it first!).

Here is one possible ordering of the deck:

Of the ten choices available for the first key card, the first nine lead to the 3 of Spades as the final key card: only the tenth, the 10 of Spades, leads to the 10 of Diamonds instead. Even more surprising is how soon most of the streams come together: seven of them converge at the 9 of Clubs in the third row.

**Mathematics:**
The principle at work here can be explained in terms of Markov processes, and
was discovered by Princeton physicist Martin Kruskal. We recommend
an online
discussion of the mathematics and
a computer
simulation, both related to the paper
``On Kruskal's
Principle'' by Wayne Haga & Sinai Robins (from *Organic Mathematics
(Burnaby, BC, 1995)*, Canadian Math Society Conference Proceedings 20
(AMS, 1997).

Here is a heuristic argument suggested by Tony Phillips which sheds some
light on the phenomenon: ``There is a 0.1 chance of the magician picking
the same first
card as the victim. That first card is number 5.5 on the average.
Counting the royal cards as 5, the average length of subsequent
steps is (1+2+...+10+5+5+5)/13=5.38, so the density of the
victim's key cards can be expected to be 1/5.38=0.186. Similarly,
the magician can be expected to choose 8 key cards
after the first, since (52-5.5)/5.38=8.64.
Each of these cards has 0.186 chance of being one of the
victim's key cards, and 1-0.186=0.814 chance of missing.
The probability of the magician not hitting one of the
victim's key cards on the first or on any of the later
choices is therefore 0.9x(0.814)^{8}=0.174, so the probability
of a win for the magician, by this rough estimate, is 0.826.''

**Source:**
This one was first learned from Gardner's *Penrose Tiles to Trapdoor
Ciphers* (W.H. Freeman & Co, 1989).

**Bonus points:**
Another possibility which increases the chance of success, which was pointed
out to us by Gordon Bean, is to spell out one card for the each letter in the
*name* of key cards, so that, for instance, ``ten'' advances three cards
(``T-E-N''). A word of caution, however: spelling values on this scale is
risky, as it's so easy for one or both of you to get confused and revert to
ordinary counting.

The implications of this trick are very interesting: are some apparently
unrelated chains of unfolding events doomed to lock together in sync after
a while? See ``An
old card trick and new Biblical hoax'' -- excerpted from *Once Upon a
Number - The Hidden Mathematical Logic of Stories* (Basic Books, 1998) by
John Allen Paulos, for musings along these lines.

It takes patience and care on the part of two people for this phenomenon to be
demonstrated correctly: that you and your victim can float merrily downstream,
oblivious to each other's progress, only to discover (with high proability) that
you are on tributaries of the same river. That being the case, may we recommend
playing some appropriately aquatic music, such as Smetana's *Ultava* -- an
orchestral tone poem which follows the course of a river from mountain streams
to its inevitable destination -- as you await the harmonious confluence!

*Revised 10/26/00*

- 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.