Four Shuffles Suffice to Randomize the Deck in Many Card Games

November 14, 2008

In many card games, just four rather than seven riffle shuffles suffice to mix up the cards. This new result is an extension of earlier work by Persi Diaconis of Stanford University and his collaborators, which established the need for seven shuffles to thoroughly randomize a standard deck.

The new wrinkle is that many gambling games ignore certain features of playing cards. In blackjack, for example, suits don't count; only card values matter. In such situations, fewer shuffles produce the requisite randomness.

Diaconis, Sami Assaf, and K. Soundararajan report their results in the paper "A Rule of Thumb for Riffle Shuffling."

Assaf had gotten the ball rolling when she experimented with small decks of cards and guessed a formula for how mixed any pack of cards might be—for whatever variables of interest. But the formula was messy. "We couldn't actually calculate" solutions, she told Science News. "We would have had to run the computer for 64 years or something like that."

Assaf turned to Diaconis and Soundararajan, who discovered an easy-to-compute equation that approximates the answer. "We found a beautiful simple pattern," Diaconis said.

Applied to various card games and any number of cards, the new equation produces quick estimates of the number of riffle shuffles required to mix up the cards. For certain games, gamblers find they need fewer than the ideal of seven shuffles and so have more time to place bets and play—and lose their money.

Casino operators aren't the only ones who stand to benefit from this discovery. The new card-shuffling result may have worthwhile applications in Monte Carlo calculations based on Markov chains, leading to significant savings in computation time.

Source: Science News, Nov. 7, 2008.