In Tetracycles: a SET Deck Magic Trick

16 April 2021 | Math Horizons www.maa.org/mathhorizons ach card in a SET deck has four features—number, color, shading, and shape—each with three possibilities so that a full deck consists of 81 cards. A magician produces a full SET deck and cuts the deck repeatedly. Four volunteers from the audience receive four consecutive cards from the magician. The magician then allows the volunteers to pick one of the four card features; let’s say they choose shape. The magician says, “Because you picked shape, would all of you holding a diamond please raise your hand?” After volunteers raise their hands accordingly, the magician asks “Now, would everyone with an oval please raise your hand?” At this point, the magician accurately lists out the four cards in order, including their numbers, colors, shadings, and shapes. The secret to our trick, which we call In TetraCycles, comes from the fact that we can arrange the deck in a very special order, in which every possible combination of four consecutive features occurs exactly once. For example, the sequence 0110 contains 01, 11, and 10. If we consider it as a cycle, which wraps around at the end, then it also contains 00. Thus, 0110 contains all four binary strings of length two. To get a sense for this property of the deck, examine fi gure 1. Such an astonishing deck arrangement exists because of the combinatorial magic of de Bruijn sequences, a staple of mathematical magic tricks. In this article, we will not talk about how to play the game of SET, as there are many great resources explaining the rules (see, for example, “Projectivizing SET” from the April 2020 issue of this magazine as well as other articles in the February 2007 and April 2017 issues). We only use the combinatorial structure of the SET deck.