Magic card trick analysis

Abstract

The magical "Cheney card trick" will be presented in this thesis using a mathematical approach. This trick is performed by a magician and an assistant in the following way: the magician leaves the room, and the assistant lets the audience draw 5 cards from a standard deck of 52 cards. Then, the assistant returns one of the 5 drawn cards to the audience and places the remaining 4 cards on the table. The magician can enter the room once the 4 cards have been placed on the table. The magician looks at the 4 cards on the table and, by these 4 cards, identifies the card that was given to the audience.
This thesis explores the underlying mathematical principles. The trick’s workings will first be explained, after which an algorithm for performing the trick will be given. In the analysis of the trick, we noticed that the size of the deck that we performed the trick with could be expanded. Given that we draw n cards from a deck of size d, we introduce an upper bound. This upper bound on the deck size is: d ≤ n! + n − 1. Furthermore, we introduce Birkhoff-von- Neumann’s theorem and Hall’s marriage theorem. Using these two theorems, we will prove that a convention for the magician and the assistant to perform the trick always exists when we attain the introduced upper bound. We will prove the existence of a convention and provide algorithms to perform the trick with a deck of cards attaining the introduced upper bound. While performing the trick, a specific order of the cards appears more often than others. When analysed, we find that a less preferable order of the cards is only necessary for approximately 10 per cent of the drawn hands. Additionally, a theorem of Gale-Shapley about stable and optimal matches is introduced, adding an extra dimension to Hall’s marriage theorem. Gale-Shapley’s algorithm is then introduced, and the possibility of finding a stable match between hands and messages is explored. Altogether, this thesis aims to find an answer to the question: What mathematical principles and ideas underlie the "Cheney card trick"?