How Many Times Should You Shuffle a Deck of Cards?
It takes just seven ordinary, imperfect shuffles to mix a deck of cards
thoroughly, researchers have found. Fewer are not enough and more do not
significantly improve the mixing.
The mathematical proof, discovered after studies of results from elaborate
computer calculations and careful observation of card games, confirms the
intuition of many gamblers, bridge enthusiasts and casual players that most
shuffling is inadequate.
The finding has implications for everyone who plays cards and everyone, from
casino operators to magicians, who has a stake in knowing whether a shuffle is
random.
The mathematical problem was complicated because of the immense number of
possible ways the cards in a deck can be arranged; any of 52 could be first in
the deck, any of 51 could be second, 50 could be third and so on. Multiplied
out, the number of possible permutations, 52 factorial, or 52;51;50, etc. is
1063 or 10 with 62 zeros after it.
No one expected that the shuffling problem would have a simple answer, said
Dr. Dave Bayer, a mathematician and computer scientist at Columbia who is a
co-author of the recent discovery. Other problems in statistics, like analyzing
speech patterns to identify speakers, might be amenable to similar approaches,
he said.
The new result ”definitely solves the problem,” said Dr. David Aldous, a
statistician at the University of California at Berkeley. ”All their
calculations are right. It’s a fascinating result.” Dr. Persi Diaconis, a
mathematician and statistician at Harvard University who is the other author
of the discovery, said the methods used are already helping mathematicians
analyze problems in abstract mathematics that have nothing to do with shuffling
or with any known real-world phenomena.
Dr. Diaconis, who is also a magician, has invented numerous card tricks and
has been carefully watching casino dealers and casual card players shuffle for
the past 20 years. The usual shuffling produces a card order that ”is far from
random,” Dr. Diaconis said. ”Most people shuffle cards three or four times.
Five times is considered excessive.”
The realization that most shuffled decks are not actually random allows
gamblers to improve their odds of winning. ”There are people who go to casinos
and make money on this,” Dr. Diaconis said. ”I know people who are out there
doing that now.”
How Casinos Do It
In Las Vegas, cards are shuffled from four to seven times, at the discretion
of the casino owners, said Richard Ingram, a Las Vegas enforcement agent for the
state gambling control board. Dr. Diaconis said he almost never sees a dealer
shuffle seven times. He said his research also shows that when dealers shuffle
several decks at once, they need to shuffle more. Two decks should be shuffled
nine times, he said, and six decks should be shuffled 12 times, which is unheard
of in the casinos.
At Trump Plaza in Atlantic City, blackjack dealers shuffle eight decks twice
at the beginning of each game, said Howard Dreitzer, who is senior vice
president of casino operations. ”We’ve tested these shuffles and feel that they
are random,” he said, adding that ”no one has ever complained.”
Bridge players usually shuffle about four times, except in some tournaments
when a computer randomly mixes the cards, said Edgar Kaplan, who is editor and
publisher of Bridge World magazine. Asked whether he expected bridge players to
change their shuffling habits, Mr. Kaplan replied, ”There will be a few who
will be affected and will doggedly shuffle seven times to the irritation of
everyone else.” As for himself, Mr. Kaplan said, ”I probably will move up from
four to five” shuffles, a decision which, the research shows, will not
appreciably improve the randomness of the shuffled cards.
Dr. Diaconis has found that many bridge players take advantage of the
non-randomness of seemingly shuffled cards. He said a bridge club in New York
State once consulted him, as a magician, to find out whether several players
were cheating. After watching play ”and doing a little thinking in between,”
Dr. Diaconis knew what was going on. These players had figured out that the
cards were not being randomly shuffled, and that they could predict the
distributions of cards by knowing what the deck looked like at the end of the
previous hand.
A Punishment of Sorts
The players ”admitted to it readily,” Dr. Diaconis said. ”But they didn’t
think they were doing anything wrong. After all, they were just thinking.” The
club asked those players not to play together for a year.
When computers were introduced into tournament bridge about 18 years ago,
some players were puzzled and others outraged by the random hands the computer
dealt and complained that the computers were not working right.
At about the same time, a bridge encyclopedia was published. The encyclopedia
”used a computer to figure out odds,” Dr. Diaconis said. ”For example, given
that between my opponents there are seven hearts, what’s the chances that one
has four hearts and the other has three? Some of these odds were at variance
with expert play. The experts had intuited – correctly – the actual ways the
cards were shuffled. People thought the encyclopedia was wrong.”
By saying that the deck is completely mixed after seven shuffles, Dr.
Diaconis and Dr. Bayer mean that every arrangement of the 52 cards is equally
likely or that any card is as likely to be in one place as in another.
The cards do get more and more randomly mixed if a person keeps on shuffling
more than seven times, but seven shuffles is a transition point, the first time
that randomness is close. Additional shuffles do not appreciably alter things.
Grist for Magicians
Magicicans have long taken advantage of the nonrandomnesss of most card
shuffling, Dr. Diaconis said. In fact, he said, Charles T. Jordan, a magician,
chicken farmer and professional contest entrant from Petaluma, Calif., made a
fair amount of money around the turn of the century by selling a card trick
exploiting the fact Dr. Diaconis said he first began to think about the
shuffling problem 20 years ago after a visit to A.T.&T Bell Laboratories in
Murray Hill, N.J. Mathematicians there told him about the problem but said they
had given up trying to solve it in 1955 because there were so many ways to
arrange a deck.
Dr. Diaconis began working with Dr. Jim Reeds at Bell Laboratories and showed
that a deck is perfectly mixed if it is shuffled between 5 and 20 times.
Next, Dr. Diaconis worked with Dr. Aldous and showed that it takes 5 to 12
shuffles to perfectly mix a deck. But, said Dr. Diaconis, ”nobody in practice
shuffles 12 times,” adding, ”We needed some new ideas.”
In the meantime, he also worked on ”perfect shuffles,” those that exactly
interlace the cards. Almost no one except a magician can do perfect shuffles
every time. But Dr. Diaconis showed several years ago that if a person actually
does perfect shuffles, the cards would never be thoroughly mixed. He derived a
mathematical proof showing that if a deck is perfectly shuffled eight times, the
cards will be in the same order as they were before the shuffling.
To find out how many ordinary shuffles were necessary to mix a deck, Dr.
Diaconis and Dr. Bayer watched players shuffle. He also watched Las Vegas
dealers to see how perfectly they would interlace the cards they shuffled.
Observations During Poker
Dr. Bayer said he seized every opportunity to get data. ”I asked everyone in
my poker game, once they dropped out of a hand, to shuffle for me,” he said.
Then the researchers did extensive simulations of shuffling on a computer. To
get the proof, the researchers looked at a lot of shuffles, guessed that the
answer is seven, and finally proved it by finding an abstract way to describe
what happens when cards are shuffled.
”When you take an honest description of something realistic and try to write
it out in mathematics, usually it’s a mess,” Dr. Diaconis said. ”We were lucky
that the formula fit the real problem. That is just miraculous, somehow.”
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Tags: bridge, cards, cards game, casino, deck, Games, How Many, magician, player, poker, proof, random, shuffle
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