After eight perfect riffle shuffles, the original order of 52 cards would be restored, provided the deck is divided exactly in half and the bottom card drops first.
Even casual gamblers understand that a deck of cards must be thoroughly shuffled to ensure everyone has an equal chance of winning based on the luck of the draw.
But how many shuffles is enough? One or two doesn’t seem adequate, but what about three, four or even five? Surely, that’s more than enough.
If you think so, fold your hand now.
The number of possible sequences in a standard deck of 52 cards is more than 8 x 10 to the 67th power, or eight followed by 67 zeros. Shuffling enough to ensure each sequence is equally possible thwarts players who might attempt to exploit discernible patterns that hang on after the conclusion of a previous hand in a poorly shuffled deck.
A century ago, a California chicken farmer was among the first to notice the advantage.
Charles Jordan would ask someone to shuffle and cut a fresh deck of cards a couple of times, then instruct the person to look at the top card, memorize it, insert it into the middle of the deck, and shuffle and cut the deck a final time. Mr. Jordan would then lay the cards face up and pluck the chosen one from the array. (Some versions have the card plucked from the middle and placed on the top or bottom of the deck, but the effect would be the same.)
Because the deck was fresh, it started out in numerical order by suit. A few shuffles weren’t enough to randomize the cards, and once they were laid out, a series of rising sequences would be visible. For example, the arrangement Ace, 5, 2, 3, 6, 7, 4 has two rising sequences interleaved: Ace, 2, 3, 4 and 5, 6, 7.
The secret of Mr. Jordan’s trick—which works about 84% of the time with three shuffles of a fresh deck, according to mathematicians Persi Diaconis and Dave Bayer—is that the chosen card, having been inserted into the middle of the deck, is now out of sequence. Mr. Jordan would simply trace the order of each suit to determine which card was out of place.
Most casual cards players under-shuffle, but it wasn’t until 1989 that Dr. Diaconis and Dr. Bayer proved there was a magic number of shuffles, what they called a cutoff, to randomize a deck.
For a deck of 52, they determined the cutoff is seven shuffles.
“If you shuffle less, the deck is far from random,” said Dr. Diaconis, who devotes two weeks to the mathematics of shuffling in a course he teaches at Stanford University. “If you shuffle a little more, it’s as close to random as can be.”
To understand what happens when the threshold is reached, imagine making marble cake with chocolate and vanilla batter.
“If you stir it a bit, you still see brown and white,” said Dr. Bayer, who teaches at Barnard College. “If you stir it more, you get tighter swirls, but you can still see both colors. At some point, it magically transitions to tan.”
Kneading dough also offers an apt illustration. If you fold the dough over once, you get two distinct layers. Fold it a second time and you get four. But if you fold it 10 times, you get 1,024 mingled layers.
In the case of randomizing cards, there is one important caveat: The seven-shuffle cutoff applies to an ordinary, and imperfect, riffle shuffle of 52 cards.
A riffle shuffle is when you divide a deck of cards in half, apply pressure to the back of the stacks with your forefinger and riffle each side with your thumbs to drop a few cards at a time onto a common pile. The number of cards that falls alternately from each side varies, and that imperfection helps ensure the randomness.
In contrast, a perfect riffle shuffle, with exactly one card dropping from each side until all the cards are down, wouldn’t randomize the deck, and after eight perfect shuffles, the original order of 52 cards would be restored, provided the deck is divided exactly in half and the card that was originally on the bottom drops first.
In the business world, casinos have the clearest interest in thoroughly shuffling cards—but they also have a competing concern.
“We consider productivity loss versus the associated risk of not shuffling completely randomly,” said Jason Sides, vice president of casino operations for the Golden Nugget in Las Vegas.
In other words, casinos want to minimize the amount of time dealers spend shuffling cards in order to maximize the amount of time customers spend gambling.
To facilitate this, casinos establish protocols for shuffling decks of different sizes, set time limits for the shuffles and conduct audits to ensure dealers adhere to the standards.
At the Golden Nugget, shuffling a single deck takes 18 seconds. A double deck takes 45 seconds. Six takes 120 seconds. And eight takes 150 seconds.
According to the mathematicians, machines don’t do a better job of shuffling cards, but Mr. Sides said they do have a benefit: “There is no downtime for shuffling.”
The odds are always with the house. Shuffle management is just one more ace in the hole.