Today, we count to twenty-one. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created them.

There's nothing quite like the inside of a casino – lights flashing, bells ringing, scantily clad cocktail servers. And, it should never be forgotten, the ever present flow of money.

Casinos are big business in the United States. They tallied thirty-two billion dollars of gambling revenue in 2006. And they really are business. Gone are the days when pit bosses kept a look out for lucky streaks. Lucky streaks are statistical aberrations. They'll wash out when all the money's counted at day's end. Today's casino operators are mathematicians.

Roulette, craps, slot machines – everything's designed to favor the house. But casinos've put a joker in their own deck. It's called blackjack. Blackjack, or twenty-one, is the only casino game where players can turn the odds in their favor.

The big difference between games like roulette and blackjack lies in the notion of independence. I spin the roulette wheel. The ball stops on a number. I spin it again. It stops on another number. But the first spin doesn't have any influence on the second. We say the spins are independent.

Blackjack's not the same. After a hand is played, some of the cards are removed from the deck. And here's the secret. When those cards are removed, the proportion of twos and sixes and kings and all the other cards changes. Sometimes, the remaining cards give players an advantage – if they know how to play their cards right.

The trick is to know when the remaining cards favor the players. Players can't possibly keep track of the many different cards that are played. So they reduce all that information to just one number. When a good card is dealt, they count up. A bad card, and they count down. That's where the name card counting comes from.

Card	2	3	4	5	6	7	8	9	10	J	Q	K	A
Count	+1	+1	+1	+1	+1	0	0	0	-1	-1	-1	-1	-1

A simple card counting strategy. Starting from zero, whenever players observe a given card they change the count by the indicated amount. When the count is positive, the deck favors players, and they should increase the size of their wager. The larger the count, the more favorable the deck.

Many people think card counting is sinister. But it's actually just good mathematics. And it all started with a young mathematician working at the MIT Computation Center. In 1961, Dr. Edward Thorp published a paper in the Proceedings of the National Academy of Sciences. It was called A Favorable Strategy for Twenty-One. He followed it up two years later with the best seller Beat the Dealer – a classic reference.

Card counting has a distinguished pedigree. But is it legal? It depends on who you talk to. Casinos need to make money. So they're concerned by players who know how to win. And that means card counters aren't welcome at casinos. But card counters are only using their skill and intelligence to turn the tables. That's something we admire in most games – not something illegal.

But there's an even bigger issue than legality. Is it worth the time and effort to count cards? The very best card counters get at most a one percent advantage over casinos. That means they need to bet a hundred dollars to make just one – on average. The reality is that players' fortunes can swing wildly. And the casinos are always on the lookout for card counters. Can you become a millionaire playing blackjack? Don't count on it.

I'm Andy Boyd at the University of Houston, where we're interested in the way inventive minds work.

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Card counting systems for blackjack have been developed over more than a half century. There are a multitude of good books that explain the fundamentals of counting and present counting systems. Among the many books:

Peter Griffin. The Theory of Blackjack, 6th ed. Las Vegas: Huntington Press, 1999.

Edward O. Thorp. Beat the Dealer: A Winning Strategy for the Game of Twenty-One. 2nd ed. New York: Vintage, 1966/1962. 

There are also many web sites that describe card counting systems and provide software with which to practice. See, for example,
Blackjack Counter for Pocket PC. Accessed April 17, 2008. 
Card Counting Strategies. Accessed April 17, 2008. 

For a popular account of a team of card counters, see:
Ben Mezrich. Bringing Down the House: The Inside Story of Six MIT Students Who Took Vegas for Millions. New York: Free Press, 2002.

Additional References:

Edward Thorp. 1961. A Favorable Strategy for Twenty-One. Proceedings of the National Academy of Sciences 47(1): 110-112.

Photo of basic strategy for blackjack from Google