I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to choose two integers between 1 and 50 and add them. Then add the largest two of the three integers at hand. Then add the largest two again. Repeat this around ten times. Alice tells the magician her final number $n$. The magician then tells Alice the next number. This is done by computing $(1.61803398\cdots) n$ and rounding to the nearest integer. The explanation is beyond the comprehension of a random mathematical layperson, but for a mathematician it is not very deep. Can anyone do better?
"The best card trick", an article by Michael Kleber.  Here is the opening paragraph:

"You, my friend, are about to witness
the best card trick there is.
Here, take this ordinary deck of cards,
and draw a hand of five cards from
it. Choose them deliberately or randomly,
whichever you prefer--but do
not show them to me! Show them instead
to my lovely assistant, who will
now give me four of them: the 7 of spades,
then the Q of hearts, the 8 of clubs, the 3 of diamonds. 
There is one card left in your hand, known
only to you and my assistant. And the
hidden card, my friend, is the K of spades."
