A magician-turned-mathematician has exposed a flaw in a casino card-shuffling machine, demonstrating how a simple guessing strategy could give gamblers a significant advantage. Persi Diaconis, a Stanford University professor and leading expert on card shuffling, was hired by a manufacturer of shuffling machines after a gang used a hidden camera to hack an earlier model.
Diaconis and statistician Susan Holmes examined a prototype of a new 'black box' shuffler designed to prevent such attacks. Although the machine appeared to randomise the deck, the mathematicians discovered that the resulting card order still contained rising and falling sequences. They devised a strategy: if the first card was the five of hearts, they guessed the next would be the six of hearts, assuming a rising sequence; if the next card was lower, they switched to guessing a falling sequence.
Using this method, they correctly predicted nine or ten cards per deck – about one-fifth of the total – enough to double or triple a card-counter's edge. Card counting, which tracks dealt cards to predict probabilities, is legal in some games but banned in casino blackjack, especially with technological assistance.
The executives were dismayed by the findings. '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 shelved the prototype and switched to a different machine.
A standard 52-card deck has 52! possible arrangements – a 68-digit number comparable to the number of atoms in the galaxy. Diaconis's work highlights the mathematical challenges of achieving true randomness in mechanical shuffling.
