It has been the
boast of Nevada's gambling casino operators that the one thing certain about a
system player is that he will lose his money systematically. "Tell us about
a man with a system and we'll send a taxi anywhere to get him," they have
said. Maybe so, but they are not sending any taxis these days to pick up Dr.
Edward O. Thorp, a 31-year-old associate professor of mathematics at New Mexico
State University. It has been a little more than a year since Thorp published a
book entitled Beat the Dealer, in which he offered a system for winning at one
of the oldest and most popular of gambling games, blackjack. Since then, says
Thorp, he and his disciples have had certain Reno and Las Vegas blackjack games
in effect barred to them, have had betting rules changed against them, have
resorted to disguises in efforts—usually successful—to get back into games in
casinos where they were known, have been blatantly cheated by casino dealers
and, most interesting of all, have been consistent winners. For the first time
in history one of the world's great gambling games has been effectively broken.
Thorp and a computer did it, and Nevada's casinos, which gross $50 million to
$70 million a year on blackjack—it is the most popular game in Reno, and second
most popular in Las Vegas, where craps ranks first—are reluctantly becoming
aware of it.
variously known as twentyone, pontoon, vingt-et-un and vanjohn, is a card game
involving a dealer and any number of players up to seven or so. The object is
to get the dealer to deal you cards totaling 21 or as near below that total as
possible, while he fails to do as well for himself. Face cards count as 10,
aces optionally as one or 11, other cards at their face value. Ordinarily, one
deck of 52 cards is used. It is continuously dealt from until it is finished,
then shuffled up and used again. You place your bet before getting one card
face down and one face up. Subsequent cards are dealt face up. If you go over
21 you "bust" and the dealer takes your money. If you stand pat and the
dealer cannot come closer to 21 than you have, you win. If you get an ace and a
10-value card you have "blackjack," an automatic 21, and the dealer
pays you one and a half times your bet unless he gets blackjack too, in which
case all bets are off. There are refinements—"doubling down,"
"insurance," "splitting pairs"—but the basic game is as stated.
When played in the normal fashion, the odds in blackjack favor the dealer by a
slim but inevitable 2% to 5%. This is the game that Thorp decided to attack
with a computer.
Thorp, who holds
a master's degree in physics and a doctorate in mathematics from UCLA, was
already a specialist in the highly abstract field of functional analysis when
he became interested in the laws of probability with respect to gambling games.
In 1958, while still at UCLA, he was looking for a vacation spot for himself
and his wife, Vivian. The lavish shows, low-cost food and nongambling come-on
lured him to Las Vegas. He had no intention of gambling while there, and Las
Vegas dearly wishes to this day that he had stuck by his decision.
Just before Thorp
left Los Angeles, a UCLA faculty friend happened to show him an article from
the Journal of the American Statistical Association. It was a report by four
Army technicians who had worked three years with ordinary desk calculators to
prove that the "house edge" in blackjack could fall as low as .0032,
virtually even money. Thorp, who knew that statistically the house edge for
craps is as high as 16%, for roulette at least 5.26%, for Nevada baccarat
1.06%, and for slot machines anywhere up to 80%, depending on the greed of the
casino owner, was intrigued by these blackjack findings and the strategy they
suggested. He wondered if there were not certain situations in which blackjack
odds actually favored the player. He could not resist trying the game. When he
got to Las Vegas he bought 10 silver dollars.
With a little
white strategy card in his hand—casinos tolerate them, but view them as a
nuisance because they, as one operator puts it, "prolong the natural
transfer of cash from the player to the house"—Thorp soon attracted a
friendly group of watchers who made patronizing cracks at the way he was
betting. After 20 minutes, during which everyone else at the table was losing,
he still had his $10. In 15 more minutes his embryonic system fell apart. But
he had been blooded.
the Thorps moved to Boston, where Ed Thorp joined the faculty of the
Massachusetts Institute of Technology. There he did what players with the same
system idea had never been able to do before. It is obvious that blackjack's
odds must change as cards get used out of the deck, but how much do they change
for every combination of cards? Thorp put the whole problem into an IBM 704,
which in three hours of electronic thinking did what it would have taken those
four Army men 2,500 man-years to do at their desks. The answers he got back
were most surprising. Thorp analyzed the results and delivered his preliminary
findings in a paper, called "Fortune's Formula," at the annual meeting
of the American Mathematical Association in Washington in 1961. At the
conclusion of his talk, 300 mathematicians rushed up and snatched mimeographed
copies of the paper as if they had a plane for Las Vegas already warming up.
The Associated Press picked up the "Fortune's Formula" story, and Thorp
and his wife were suddenly so besieged with phone calls and letters from people
who wanted to join them on a we'll-pay-the-bills vacation in Nevada that they
had to leave their Cambridge flat temporarily to get some quiet, and to
consider their next move.
Thorp turned down
all offers to back him except one. It came from two affluent big-time gamblers
who were willing to risk $100,000. The unlikely trio was soon on its way to
Reno and Lake Tahoe, where Thorp's horn-rimmed glasses, dark hair and fresh,
scrubbed face hardly struck terror into the pit bosses. Then one night, after
several days of careful and uneventful testing, Thorp sat down at a Lake Tahoe
table where blackjack was being dealt by a pretty girl and began tossing $25
and $50 chips into the game. Deliberately, he appeared annoyed at drifters who
slowed up the game by making dollar bets, and soon an obliging pit boss, who
was sure he had a pigeon about to lay a golden egg, put up a discreet sign
reading "Minimum Bet: $25." That speeded up play. In 12 minutes Thorp
won a few hundred dollars. The pit boss, a little less chipper, was relieved to
see what he thought was an older, affluent sucker join the table. But it was
one of Thorp's backers, and Thorp was quietly but firmly calling the signals.
In 30 minutes the pair broke the table's bank. Dealers were switched, the money
tray behind the table was refilled and a small crowd came in close to watch,
for a betting system that seems to work is a rare joy to behold in Nevada. Free
money always is.
The two men
played preposterously by most normal standards. They drew cards and risked an
almost certain bust when any reasonable player would have stood pat. They split
pairs and they doubled their bets when the chances against them seemed
staggering. Two hours and $17,000 later, they broke the bank again. That was
about all there was to it. The two wealthy backers returned to one of those
strange places that wealthy backers spring from, and Thorp returned to MIT to
write Beat the Dealer.
Did their venture
herald the end of blackjack? Eventually—perhaps. Immediately—no. Of the three
million blackjack players who pass through Nevada in a year, not more than
50,000 have read Beat the Dealer ( Random House, $4.95), even though it is the
most requested book in the Las Vegas Public Library. Of these 50,000, only a
tiny percentage have the determination to master the strategies explained
there—surprising, since Thorp estimates a mere 20 hours of study and practice
is all that is needed for the average card player to become very
Here is how
Thorp's system works. Unlike craps or roulette, which mathematicians call
"independent trials processes" because one play is unrelated to the
next, blackjack hands are highly dependent upon what has gone before. As the
deck is used up and certain cards are played, the proportion of favorable to
unfavorable cards left in the deck is changeable. The same complex formula that
says flatly that there is no honest way to beat the house permanently at craps
or roulette states just as categorically that blackjack, played properly, can
actually favor the player. Only poor tactics, poor money management or house
cheating will enable the house to keep its edge.