How a Mathematician-Magician Revealed a Casino Loophole

It's the tale of a company manufacuring precision card-shuffling machines for casinos — and a gang of hustlers who used a hidden video camera to film the shuffler's insides. "The images, transmitted to an accomplice outside in the casino parking lot, were played back in slow motion to figure out the sequence of cards in the deck," remembers the BBC, "which was then communicated back to the gamblers inside. The casino lost millions of dollars before the gang were finally caught." So the company turned for help to a mathematician/magician: The executives were determined not to be hacked again. They had developed a prototype of a sophisticated new shuffling machine, this time enclosed in an opaque box. Their engineers assured them that the machine would sufficiently randomise a deck of cards with one pass through the device, reducing the time between hands while also beating card-counters and crooked dealers. But they needed to be sure that their machine properly shuffled the deck. They needed Persi Diaconis. Diaconis, a magician-turned-mathematician at Stanford University, is regarded as the world's foremost expert on the mathematics of card shuffling. Throughout the surprisingly large scholarly literature on the topic, his name keeps popping up like the ace of spades in a magician's sleight-of-hand trick. So, when the company executives contacted him and offered to let him see the inner workings of their machine — a literal "black box" — he couldn't believe his luck. With his collaborator Susan Holmes, a statistician at Stanford, Diaconis travelled to the company's Las Vegas showroom to examine a prototype of their new machine. The pair soon discovered a flaw. Although the mechanical shuffling action appeared random, the mathematicians noticed that the resulting deck still had rising and falling sequences, which meant that they could make predictions about the card order. To prove this to the company executives, Diaconis and Holmes devised a simple technique for guessing which card would be turned over next. If the first card flipped was the five of hearts, say, they guessed that the next card was the six of hearts, on the assumption that the sequence was rising. If the next card was actually lower — a four of hearts, for instance — this meant they were in a falling sequence, and their next guess was the three of hearts. With this simple strategy, the mathematicians were able to correctly guess nine or 10 cards per deck — one-fifth of the total — enough to double or triple the advantage of a competent card-counter.... The executives were horrified. "We are not pleased with your conclusions," they wrote to Diaconis, "but we believe them and that's what we hired you for." The company quietly shelved the prototype and switched to a different machine. The article also explains why seven shuffles "is just as close to random as can be" — rendering further shuffling largely ineffective.

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