A magician at the International Congress of Mathematicians

Persi Diaconis was a well-known magician in the United States. Today he is a professor of mathematics. He uses his research to invent new magic tricks.

"If it was possible to be a professor of magic I would never have become a mathematician or statistician", says Persi Diaconis, Professor of Statistics at the famous Stanford University in Palo Alto (California). On Thursday he will give the final plenary lecture at the International Congress of Mathematicians, which has been guest in Berlin over the past ten days. What his academic title and reputation as a researcher do not disclose: Persi Diaconis really is a magician, and so famous in America that he would have become a professor of magic long ago if such a title existed.

Things all began in 1959, when Diaconis met the legendary Dai Vernon in a New York magic café. Vernon was so impressed by the 14-year-old Diaconis that he asked him if he would like to join his act. Two days later, Diaconis left home without saying farewell. Only ten years later did he turn up again to finish college by taking evening classes. Five years later he had obtained a doctorate at Harvard University. In the same year he was made professor at Stanford.

"Actually, I am better at magic than at mathematics", says Diaconis, "but magic gets very monotonous when you are doing it professionally. You are always having to do the same tricks in order not to disappoint the public." Today he only rarely appears on stage, although he still develops tricks for other magicians. But the ten years he spent as a magician on street corners, in clubs, hotels and enchanting high society have left their mark on the way he does mathematics. "Things must not get too abstract for me. I prefer to puzzle over real problems", he says. The reality he is referring to, however, has nothing to do with technology and industry, but with his passion - magic. Among other things, Diaconis has researched the mixing of cards, card tricks, and fair dice. In the process, he has discovered new mathematical truths, closely linked to existing theories in algebra and probability theory. He has also made significant contributions to mathematical statistics. For example, he has investigated the circumstances under which it is possible to derive reliable conclusions from large amounts of data material. He was able to prove that under certain circumstances the interpretation of data remains unreliable, even if more and more numerical material is added to the statistics.

Here are three popular examples of the research of Persi Diaconis:

Shuffling cards
How often is it necessary to shuffle a pack of 52 cards until the order of the cards is really random? About seven times is the mathematical answer - but only if the cards are mixed properly each time and there is no cheating. The formula that Persi Diaconis developed together with David Bayer of Columbia University (New York) is 1.5 x log2n, where n is the number of cards. The central argument in the proof of this formula is that after shuffling a certain number of times the original order has been completely lost. Exactly when this happens depends on the type of shuffling and is not easy to calculate. The mathematics involved comes from probability theory, and the models used to describe the zigzag Brownian motion of molecules in a liquid.

Fair dice
A normal dice has six sides. But what would a dice have to look like if it had seven, eight, nine, ten, or more sides. At first one thinks of symmetrical objects: All faces must be of the same size and all angles between the faces must be identical - like normal dice. Indeed, all regular polyhedrons are fair dice, such as for example the tetrahedron (with four equal triangular sides), the octahedron (a double-pyramid with eight triangular sides) or the dodecahedron (12 sides). But there are also fair dice which are not completely symmetrical. An example are prisms, whose sides are parallelograms with end faces that are equal, parallel polygons. Together with Joseph Keller (Stanford University), Diaconis could prove that there are no symmetrical fair dice with five sides, but there can be unsymmetrical ones which are fair. Their shape depends on the type of surface the dice are thrown on - for example sand or wood.

Guessing cards
An experiment for two players and a deck of 52 cards: The first player takes a card from the pack and holds it in their hand. The other player has to guess what it is. If the guess is wrong the first player just says "No", lays the card face down and picks up the next one. How many cards will the second player guess correctly? The mathematical answer is "nearly two" (or to be exact e - 1, where e is the exponential constant 2.718). If the cards are laid face up after an unsuccessful guess, the second player knows which cards are no longer in the pack and the number of successful guesses increases to "four to five".

Such calculations can also be used to check some claims of extra-sensory perception. And Persi Diaconis has made a name for himself in this field by using his skills as mathematician and magician to uncover charlatans. However, there is another application for his research: "Mathematics helps me to invent new magic tricks", says Diaconis. His students are often a willing audience, because every now and then Diaconis will show a few tricks in his lectures. And this is a further reason why his plenary lecture on Thursday is being awaited with some considerable anticipation. The topic: "From shuffling cards to walking around the building", 27 August, Auditorium Maximum of the TU Berlin, Straße des 17. Juni 135, Berlin-Charlottenburg, 2.00 p.m.

Vasco Alexander Schmidt