To study riffle shuffles rigorously, Diaconis used a powerful mathematical tool called a Markov chain.
“A Markov chain is any repeated action where the outcome depends only on the current state and not on how that state was reached”, explains Sami Hayes Assaf, a mathematician at the University of Southern California. This means that Markov chains have no “memory” of what came before. This is a pretty good model for shuffling cards, says Assaf. The result of the seventh shuffle depends only on the order of the cards after the sixth shuffle, not on how the deck was shuffled the five times prior to that.
Markov chains are widely used in statistics and computer science to handle sequences of random events, whether they are card shuffles or vibrating atoms or fluctuations in stock prices. In each case, the future “state” – the order of the deck, the energy of an atom, the value of a stock – depends only on what’s happening now, not what happened before.
Despite their simplicity, Markov chains can be used to make predictions about the likelihood of certain events after many iterations. Google’s PageRank algorithm, which ranks websites in their search engine results, is based on a Markov chain that models the behaviour of billions of internet users randomly clicking on web links.
Working with Dave Bayer, a mathematician at Columbia University in New York, Diaconis showed that the Markov chain describing riffle shuffles has a sharp transition from ordered to random after seven shuffles. This behaviour, known to mathematicians as a cut-off phenomenon, is a common feature of problems involving mixing. Think of stirring cream into coffee: as you stir, the cream forms thin white streaks in the black coffee before they suddenly, and irreversibly, become mixed.
Knowing which side of the cut-off a deck of cards is on – whether it is properly shuffled or if it still preserves some memory of its original order – gives gamblers a distinct advantage against the house.
In the 1990s, a group of students at Harvard and MIT were able to beat the odds playing blackjack at casinos around the US by using card counting and other methods to detect if the deck was properly shuffled. Casinos responded by introducing more sophisticated card-shuffling machines, and shuffling the deck before it is fully played, as well as stepped-up surveillance of players. But it is still rare to see a deck of cards shuffled by machine the requisite seven times at a casino.
Casino executives may not have paid much heed to Diaconis and his research, but he continues to have an enormous influence on mathematicians, statisticians and computer scientists who study randomness. At a conference held at Stanford in January 2020 to honour Diaconis’s 75th birthday, colleagues from around the world gave talks on the mathematics of genetic classification, how cereal settles in a shaking box, and, of course, card shuffling.
Diaconis doesn’t care for gambling much himself – he says there are better and more interesting ways to make a living. But he doesn’t begrudge players who try to get an edge by using their brains.
“Thinking isn’t cheating,” he says. “Thinking is thinking.”
*Shane Keating is a science writer and senior lecturer in mathematics and oceanography at the University of New South Wales, Sydney
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