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                  Copyright 1990 The New York Times Company                     
                               The New York Times                               
                 January 9, 1990, Tuesday, Late Edition - Final                 
                              Correction Appended                               
SECTION: Section C; Page 1, Column 1; Science Desk                              
LENGTH: 1349 words                                                              
HEADLINE: In Shuffling Cards, 7 Is Winning Number                               
BYLINE: By GINA KOLATA                                                          
   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  
   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.''      
CORRECTION-DATE: January 17, 1990, Wednesday, Late Edition - Final              
   An article in Science Times on Jan. 9 about card shuffling misstated the     
value of 52 factorial, or 52 X 51 X 50, etc. It is approximately 0.8 X 10 to the
68th power.                                                                     
GRAPHIC: photo: Dr.  Persi Diaconis,  Harvard mathematician, used a computer to 
analyze the shuffling patterns. (NYT/Rick Friedman)