In September 1961, a month after our test of the roulette computer in Las Vegas, Vivian, Raun, and I moved to Las Cruces, New Mexico, where I began my duties as a professor in the Mathematics Department at New Mexico State University. Then a town of thirty-seven thousand people, located in the high desert about four thousand feet above sea level, Las Cruces was established near a principal source of the state’s water, the Rio Grande. Towns were widely spaced in the desert expanse, and the nearest population center was El Paso, Texas, about forty-five miles to the south. After the University of New Mexico in Albuquerque, some two hundred miles to the north, NMSU was the next most important campus in the state university system. When I arrived it was being transformed from an agricultural college to a university. Just to the east of the campus was “A” mountain, a large hill with an enormous white A for “Aggies.” Some claimed that when the football team learned the first letter of the alphabet, it would be changed to B.
Our four years in New Mexico were memorable. Our younger daughter, Karen, was born there and our son, Jeff, was born in nearby El Paso. About twenty miles away was White Sands Proving Ground and National Monument, where we found some relief from summer heat, as the sun’s rays were efficiently reflected away by the white gypsum “sand.”
I followed up on my childhood interest in astronomy, enjoying New Mexico’s dark skies through a small telescope. The astronomical highlight was a private lunch with Las Cruces resident and fellow NMSU professor Clyde Tombaugh (1906–97), who became world famous in 1930 when, at the Lowell Observatory in Flagstaff, Arizona, he discovered the planet Pluto (recently demoted to a “dwarf planet”). My student William E. “Bill” Walden, who worked at Los Alamos, arranged for me to spend an afternoon there with Stanislaw Ulam (1909–84), one of the twentieth century’s greatest mathematicians. Ulam, part of the Manhattan Project that developed the atomic bomb, later supplied crucial ideas for the hydrogen bomb—the Ulam-Teller concept for thermonuclear weapons.
While teaching graduate courses and doing mathematical research at NMSU, I wondered whether what I had learned so far would enable me to beat other gambling games. One of the casino games I noticed on my Nevada blackjack trips was baccarat, which James Bond plays both in Ian Fleming’s book Casino Royale and in the dramatic beginning of the original version of the movie of the same name. Long played in Europe for high and sometimes unlimited stakes, this Continental favorite had been introduced in a slightly modified form by a few Las Vegas casinos. With similarities to blackjack, baccarat was a natural target for my methods. Fortunately, Bill Walden, a computer scientist with an interest in applying mathematics, was happy to be recruited. We began our analysis of baccarat in 1962, with the goal of finding to what extent we might be successful using my card counting methods.
Nevada-style baccarat was dealt from eight decks, totaling 416 cards. These have the same values as in blackjack, except only the last digit counts. Thus Aces are 1, 2s through 9s are their numerical value, and 10s, Jacks, Queens, and Kings count as 0, not 10. The game begins after the cards are shuffled and a blank “cut card” is inserted into the pack of cards faceup near the end. The 416 cards are then put into a wooden dealing box known as a shoe. The first card is exposed, its value noted, and this number of cards is discarded, or “burned.” If the exposed card is a 10 or a face card, then ten cards are burned.
A casino table had twelve seats, occupied by an assortment of customers and shills (house employees who bet money and may pretend to be players in order to attract customers). There are two main bets on the layout: Banker and Player.
After the players place their wagers, the croupier deals two cards each facedown to a betting spot on the table layout labeled BANKER and another spot called PLAYER. Then the croupier turns the hands faceup. As with individual cards, only the last digit of a total counts. For example, 9 + 9 = 18 counts as a total of 8. If the first two cards of a hand total 8 or 9, termed a natural 8 or a natural 9, as the case may be, all bets are settled without any further cards being dealt. If neither the Player spot nor the Banker spot receives a natural, each hand, starting with the Player’s, either receives one more card from the croupier or stands according to a set of rules. The high hand wins. If there is a tie the bettors get their money back.
Our analysis of baccarat followed the same approach I had used for blackjack because of the similarities between the two games. To begin, we calculated, for the first time ever, the correct values for the house advantage in the Nevada version of baccarat for the two bets, Banker and Player. For the Banker, it was 1.058 percent of all bets placed; it was 1.169 percent if ties were not included. For the Player bet it was 1.235 percent, or 1.365 percent omitting ties. These figures assume that the player does not keep track of the cards that have been used. The casino edge is different for the two bets, Banker and Player, because the rules for drawing a third card differ and also because winning bets on Banker have to pay the casino 5 percent.
What if a player does count cards?
To find out, Bill Walden and I proved what we called the Fundamental Theorem of Card Counting, which says, in a precise mathematical way, that the advantage from card counting becomes better as more cards are seen. This means the best situations come toward the end. We found that even those were tiny and rare.
The reason baccarat does not have enough opportunities is because the impact of removing one card from the pack in baccarat is about one-ninth what it is in blackjack, so the effect on the house edge is correspondingly smaller. Also, the house edge to be overcome is greater, being more than 1 percent.
However, in addition to the main bets of Banker and Player, the baccarat layout had four separate side bets: Banker natural 9, Player natural 9, Banker natural 8, and Player natural 8. Banker natural 9 won if the first two Banker cards totaled 9, in which case the bet paid 9:1, meaning that a winning $1 bet was paid a $9 profit. The other three side bets had the same payoff.
For the non-counter these bets were terrible, with a house edge of 5.10 percent for each of the two bets on natural 9 and 5.47 percent for natural 8. But we discovered that although a card counter couldn’t beat the Banker and Player bets, he could beat these side bets! As I predicted by reasoning and we verified by computation, the edge on the side bets varied widely as cards were used. About one-third of the way through the shoe, good opportunities appeared, and things got better as more cards were played.
We devised a practical card counting system, which used the fact that when the remaining cards had a big excess of 9s, the natural 9 bet favored the Player. An 8-rich deck did the same for bets on Player natural 8.
For a casino test, I recruited the chairman of the Mathematics Department, Ralph Crouch. We practiced counting down the eight-deck pack. This required a running tally of the number of unseen cards, as well as the number of 8s and 9s among them. Counting was harder than in blackjack because eight decks have 416 cards, including thirty-two 9s and thirty-two 8s, and we would track all three quantities.
Ralph was unlike any other math department chairman I have ever encountered. Of middle height, florid-faced, bubbly, and chatty, he was an extreme extrovert. This set him apart from the typical introverted mathematician. A well-known joke is, “How do you tell whether a mathematician is an introvert or an extrovert?” Answer: “If he looks at his shoes when he talks to you, he’s an introvert. If he looks at your shoes, he’s an extrovert.” The life of the party, Ralph promoted departmental get-togethers fueled by “Las Cruces Punch,” a concoction made in a huge bowl with two or more gallons of Bacardi rum along with frozen orange juice, pineapple juice, and lemonade. Vivian and I evaded as many of these parties as we could and made brief polite appearances when we did go. Years later, when my daughters came across the recipe and the proportions—mostly rum—they wondered how anyone remained vertical.
I am often asked what it takes to be a successful card counter. I’ve found that an academic understanding is not enough. You need to think quickly, be disciplined enough to follow the system, and have a suitable temperament, including the ability to switch your mind into the here and now and stay focused on the cards, the people, and your surroundings. Better still is to have an “act” or persona that makes you seem like a type of player with which casinos are familiar.
I thought Ralph would be perfect, along with his male golfing buddy Kay Hafen, the controller of the university, for my proposed baccarat team. Kay was low-key, levelheaded, and unflappable. In the practice sessions I held with them, both learned to count well. Our wives came, too, and Vivian, who hadn’t been on the various blackjack trips, was relieved to be able to monitor my safety firsthand. When we weren’t playing, the six of us planned to enjoy ourselves around town.
We drove to Las Vegas during the university’s 1963 spring break. We arrived at the Dunes shortly before their baccarat game started at 9 P.M., and acted as though we didn’t know one another. Velvet ropes separated the baccarat alcove from the rest of the casino floor. The imposing raised table had six chairs at each horseshoe-shaped end. Several female shills were already seated when I sat down. Despite my publicity from blackjack, I went unnoticed by the casino people. At least at first.
As play began, a crowd gathered outside the velvet rope to watch what could become a high-stakes game. Betting limits were $5 to $2,000 on the main bets and $5 to $100 on the side bets, equivalent to about ten times as much in 2016.
Then someone cried out, “There’s the guy who wrote the book.” The baccarat supervisor’s eyes popped open and he ran to a nearby phone. One of the wives, eavesdropping on the call upstairs, saw the man’s concern change to reassurance and then amusement. Beating blackjack is one thing; baccarat is another. Our spy heard, “Ha, ha. Let him play!” And so play we did.
Our first night was pleasant. With the newly shuffled pack of 416 cards, all bets favored the house, so I started with the smallest allowed bets on Banker, $5, while I tracked the number of 8s, 9s, and total cards that remained, and waited for favorable situations. I set the size of our big bets to give a win rate of $100 per hour, hoping that would be low enough to keep us from being barred.
It took about forty-five minutes to play through a shoe. After playing two of them, I rested while Ralph and Kay played the next one. They split the work, with Ralph monitoring the natural 8 bets, while Kay counted and bet for the natural 9s. This was easier because they each had only two separate card totals to keep track of, rather than three. After one shoe they rested and I played two more sessions. We continued this pattern. When the game shut down at the usual time of 3 A.M., we were ahead five or six hundred dollars, about what we expected.
The next night, as I sat down for the start of the game, the atmosphere had changed. The casino crew was distant and unfriendly—and the shills did something strange. On the previous night the game began with me, another player or two, and half a dozen female shills spread among the twelve seats. Soon other players were attracted to this falsely busy scene and joined the action. When all seats were filled, one shill popped up, leaving just a single seat vacant for the maximum power to draw bettors: Just one seat left—grab it while you can. As soon as a new player was drawn to that seat, yet another shill popped out of hers. This dance of shill-in, shill-out, leaving exactly one seat empty, had continued through the evening. But on this, our second night, the shills on each side of me remained parked, watching closely. At the time I had a bronchial tickle that triggered frequent strident coughs. Our undercover wives were amused when the shills assigned to me became concerned for their health, rebelled, and had to be ordered to stay at their posts.
As we continued to win, other players starred in their own human dramas. Vivian noted a bleached-blond Asian lady with long magenta fingernails. Heavily made up and bejeweled, she was betting the $2,000 limit on each hand and losing. She owned a chain of supermarkets and in a couple of hours she lost one of them. Baccarat is a game favored by big bettors. By 1995 baccarat in Nevada accounted for half as much casino profit as did blackjack, yet with only one-fiftieth as many tables. A baccarat table was twenty-five times as profitable as a blackjack table.
The game shut down again about 3 A.M. on this second night. After we counted our winnings, Ralph and Kay returned to the bar for a drink. The pit boss and a couple of men from the casino were there with the baccarat shoe and the eight decks of cards. They were muttering and examining the cards one by one, looking for bends, crimps, markings, and any other clues as to how we were winning.
The third night began with obvious hostility toward me by all the employees in the pit. They conspicuously watched my every move. To mislead them, I frequently touched my thumb behind my ear, as though I were a cheat marking the cards with “daub,” a nearly invisible Vaseline-like substance easily seen through special glasses. I hoped they would waste another night examining every card, looking for what wasn’t there. On the first two evenings they had repeatedly offered me drinks, but I had chosen coffee with cream and sugar instead. Tonight it was war and they didn’t offer me anything. We won again.
When I sat down to play on the fourth night, the atmosphere had again changed, drastically. The pit boss and his minions were smiling and relaxed. They seemed pleased to see me. Then they volunteered “coffee with cream and sugar, just the way you like it.” I was deep into the first shoe happily winning and drinking my coffee when suddenly I couldn’t think. I could no longer keep the count. I was shocked because I had managed well enough through noise, smoke, conversation, the pressure of high-speed play, the excitement of losing or winning, and the impact of alcoholic beverages. Something unexpected had taken place. I took my chips and left, replaced on the next shoe by Ralph and Kay.
The wives saw that my pupils were hugely dilated. Bellamia Hafen, who was a nurse, said that she had seen this often when people who had used drugs were admitted to her hospital. I wanted to collapse into sleep but Vivian, Isobel Crouch, and Bellamia plied me with black coffee and walked me for several hours until the effects began to wear off. Ralph and Kay played through the evening until night four ended. We won again.
After considerable discussion among us, I took my seat for the start of the game on night five. No longer smiling, the boys again offered me coffee with cream and sugar. I said, “No thanks. Just bring me a glass of water.” The rest of my group groaned silently. It took a suspiciously long time and when the water came I expected it to have something extra. To find out, I carefully put only a single drop on my tongue. Ugh! It tasted as if a box of baking soda had been emptied into the glass. But that single drop was enough to flatten me again. I wondered what a swallow would have done.
With numb brain and dilated pupils I left and repeated the black-coffee-and-walking routine. Meanwhile Ralph and Kay were asked to leave, permanently, and the same went for all of their friends.
There was one more baccarat game with the side bets and it was at the Sands casino. After a day of R&R, I went there with our bankroll and took a seat at the table. I revised our target from $100 an hour to $1,000, figuring that the Dunes would have contacted the Sands so I would soon be barred. After two and a half hours I was ahead $2,500. Then Carl Cohen, part owner of the Sands and the person in charge, paid me a table-side visit. Carl had previously disciplined Frank Sinatra for making a commotion in the casino. When Sinatra persisted, Cohen saw to it that he didn’t return, even though Sinatra was a minor part owner. It was Cohen who now told me not to play at his casino anymore. I asked why and he said, “No reason. We just don’t want you to play here.” He was accompanied by the largest security guard I had yet seen. Discussion was pointless. I left.
During our six nights of play, we had proven the system at the tables. We validated the theoretical mathematical calculations and demonstrated yet another application of the Kelly system for betting and investing. But our trip would have an unnerving postscript.
The six of us left Las Vegas the next morning to drive back to Las Cruces. I was at the wheel as we went down a mountain road in northern Arizona. We were going sixty-five miles an hour when the accelerator pedal suddenly jammed. The steep downhill and the wide-open throttle were too much for the brakes. The car sped up to eighty miles an hour and the turns in the road became unmanageable.
With little time to think and my foot pressing as hard as I could on the brakes, I also set the emergency brake, downshifted so the engine would help slow the car, and cut off power by turning off the ignition. I finally managed to stop in a turnout. A Good Samaritan who understood cars pulled over to help us. Opening the hood to see why the accelerator jammed, he found a part that had come unscrewed from a long, threaded rod, something he had never before seen happen and found baffling. He fixed it and we continued on our way, alive, relieved, and sobered.
We had proven the system worked at the tables like it did in theory. As a result, both the Dunes and the Sands removed the natural 8 and natural 9 bets.
While at New Mexico State, I invested money from book royalties and gambling winnings in stocks. But I was ignorant of the market as well as unlucky. The results were poor. I wanted to do better. Investments presented a new type of uncertainty, but the theory of probability might help me make good choices.
Things came together when I realized that there was a far greater casino than all of Nevada. Could my methods for beating games of chance give me an edge in the greatest gambling arena on earth, Wall Street? Ever curious, I decided to find out. I began to teach myself about the financial markets, lighting my way with an unusual lamp, the knowledge I had gained from gambling games.