TITLE: Magic trick based on deep mathematics
QUESTION [181 upvotes]: 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?

REPLY [5 votes]: A recent episode of Penn and Teller's Fool Us had a trick by Hans Petter Secker which exploits the parity of a permutation in a lovely way.  It may be useful the next time one has to teach the sign of a permutation!
Brief description of the trick in the video:  The magician has sent a box over.  Penn picks up a Rock from the box; Teller then picks Scissors from the box; and Alysson gets the remaining crumpled sheet of paper.  The magician (on video) says that he will predict what they'll do (in the context of the Rock-Paper-Scissors game).  He writes down a prediction, and invites Alysson to pick any two of them and swap their objects. The magician reveals his correct prediction.  Next he writes down two predictions, and asks Teller to perform a swap, and then Penn to perform a swap (I might be forgetting who selects the swaps, but two swaps are made).  Again the magician reveals his correct predictions.  Finally he gets them to make three swaps.  This time he has not written down any predictions, but when the crumpled paper is smoothed out, there are three correct predictions written on it. Enjoy!
https://www.youtube.com/watch?app=desktop&v=hf7Sy7FPal0