In their recently published book, Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks, Persi Diaconis and Ron Graham – both mathematicians and a professional magician and juggler, respectively – reveal the “secrets of amazing, fun-to-perform card tricks – and the profound mathematical ideas behind them.” At BIRS, we particularly like the chapter, Is this Stuff Actually Good for Anything, which tackles the subject of Bruijn sequences, in which the authors include a section that provides a glowing review of the BIRS workshop, Generalizations of de Bruijn Cycles and Gray Codes, held in December of 2004.

The BIRS workshop was quite likely the first meeting of its kind to focus on the theory, constructions and generalizations of de Bruijn cycles, despite the fact that many people in diverse areas have been working on aspects of universal cycles of one kind or another for quite some time. Its intent was to give an overview of the various known results on de Bruijn sequences and universal cycles in general, and to exploit the diversity of the attendees in order to stimulate new work, focussing on several questions in which progress seems imminent or likely. According to Diaconis and Graham, BIRS provided an ideal setting for such interactions.

In their article, they describe how 25 researchers, including the likes of Brendan McKay (Australian National University), Eduardo Moreno (Universidad Adolfo Ibañez, Chile), Robert Johnson (Queen Mary University of London) and international experts including Hal Fredricksen (Naval Postgraduate School) who has over 40 years of experience in the field, Frank Ruskey (University of Victoria) who “has the world’s best programs for generating all kinds of de Bruijn sequences and Gray codes on his Web site” and Carla Savage (North Carolina State University), an expert on nonstandard constructions, came together in the “exotic location” of Banff, Alberta.

They go on to state how:

“There were really friendly introductory and expository talks aimed at bringing newcomers up to speed and making sure we were all on the same page. There were announcements of new results, big and small…Much time was spent in small groups where people go over special cases slowly and ask each other “silly questions” that might be embarrassing if asked in a large group.”

They also describe “one of the most spectacular new results [which] was Robert Johnson’s solution of the notorious ‘middle-layer’ problem” and how this particular problem relates to the magic behind their card tricks, stating that “Johnson’s result introduces new ideas and techniques that will surely be of help in other graph cycle problems.”

Diaconis and Graham conclude their article with a testament to the workshop’s success, citing progress made on old conjectures and new conjectures posed, but mostly, focusing on the community that was formed as a result of the time these researchers spent at BIRS. They state, “to find others who think this small world of problems is beautiful and important made a deep impression on all of us.”

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P. DIACONIS and R. GRAHAM, Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks, Princeton University Press (2012), 42-26.