Using an ordinary blue-backed deck, place such a stack, starting with the 8♥, on top of the rest of a deck, keeping it there throughout some fair-seeming shuffling, and deal the stack into a face-down circle.
Now, explain that three people must pick adjacent cards to represent partners for a dancing Queen: ask somebody to fan through the remainder of the deck and pull out the Q♥. Have this card placed face-up in the middle of the circle of eight cards.
Turn away, while three adjacent cards in the circle are peeked at by different spectators. Have these three cards, representing dancing partners, lined up in order in front of the Queen, while the other five cards are shuffled back into the deck. Turn back, and explain that each card/partner has a net worth, which is indicated by its numerical value. Remind people that Aces are worth 1, Jacks 11, and so on, which adds to the illusion that any cards could be involved. Furthermore, red cards are trustworthy as dancing partners, for a red Queen, but black ones are not.
Have the Q♥ turned face-down too, and ask the three participants to move their cards around along with the Queen, to simulate three partners dancing with her. Say that having seen them dance, you now know their identities. Just to check, give your participants a choice: either they will reveal to you the total worth of the selected partners, by summing the three card values while you turn away again, or they will indicate which partners are trustworthy. Regardless of whether you are given the value sum or learn which of the three cards are red, say, "Just as I thought," and proceed to identify all three cards correctly.
In the first case, use the technique explained in the June 2008 Card Colm, not forgetting that the suits above differ from the ones used there. In the second case, you can deduce which part of the RRRBBBRB cycle, and hence the d4b8 bracelet, was sampled, from the red/black distribution alone.
Other FoundersThere is a considerable magic literature on applications of de Bruijn sequences, both binary and more general ones, which has been and will likely remain mostly invisible to outsiders. It is no exaggeration to say that entire books have been written on the subject, featuring innovative applications and generalizations of what we've considered. Pioneers from decades past include Bob Hummer, Larsen & Wright, Ron Wohl, Max Maven and Leo Boudreau. The roots of the subject can be traced back to Charles Jordan's "Coluria" from 1919.
Just as we have only revealed the tiniest tip of the magic iceberg above, we have also only skimmed the surface of the many fascinating mathematical possibilities discussed in the Diaconis & Graham chapter of A Lifetime of Puzzles (AK Peters), which explains magic ways to combine universal cycles, far-reaching generalizations of de Bruijn sequences first considered by Fan Chung along with Diaconis & Graham in 1992. Readers hungry for more knowledge and inspiration (as well as open problems) are strongly encouraged to start there.
Colm Mulcahy (email@example.com) completed his PhD at Cornell in 1985, under Alex F.T.W. Rosenberg. He has been in the department of mathematics at Spelman College since 1988, and writing Card Colms---the only MAA columns to actively encourage lying on a regular basis---bi-monthly since October 2004. For more on mathematical card tricks, including a guide to topics explored in previous Card Colms, see http://www.spelman.edu/~colm/cards.html.
"A Two Role," "Truer Purpose" and "Dancing Queen" are anagrams of the titles of well-known Abba songs. "Other Founders" is an anagram of "Four-Shortened."