It wasn’t the money that drew me to blackjack. Though we could certainly use extra dollars, Vivian and I expected to lead the usual low-budget academic life. What intrigued me was the possibility that merely by sitting in a room and thinking, I could figure out how to win. I was also curious to explore the world of gambling, about which I knew nothing.
Back from Las Vegas, I headed for the section in the UCLA library where the mathematical and statistical research articles were kept. Grabbing from its shelf the volume containing the article with the strategy I had played in the casino, I stood and began to read. As a mathematician, I had heard that winning systems were supposed to be impossible; I didn’t know why. I did know that the theory of probability had begun more than four hundred years earlier with a book on games of chance. Attempts to find winning systems over the following centuries stimulated the development of the theory, eventually leading to proofs that winning systems for casino gambling games were, under most circumstances, impossible. Now I benefited from my habit of checking it out for myself.
As my eyes gobbled up equations, suddenly I saw both why I could beat the game and how to prove it. I started with the fact that the strategy I had used in the casino assumed that every card had the same chance of being dealt as any other during play. This cut the casino’s edge to just 0.62 percent, the best odds of any game being offered. But I realized that the odds as the game progressed actually depended on which cards were still left in the deck and that the edge would shift as play continued, sometimes favoring the casino and sometimes the player. The player who kept track could vary his bets accordingly. With the help of a mental picture based on ideas from an advanced mathematics course, I believed the player edge must often be substantial. Moreover, and this also was new, I saw how the player could condense and use this information in actual play at the table.
I decided to begin by finding the best strategy to use when I knew which cards had already been played. Then I could bet more when the odds were in my favor and bet less otherwise. The casino would win more of the small bets, but I would win a majority of the big wagers. And if I bet enough where I had an advantage, I should eventually get ahead and stay ahead.
I left the UCLA library and went home to figure out the next steps. Almost at once, I wrote to Roger Baldwin, one of the four authors of the blackjack article, asking for details about the calculations, telling him I wished to extend the analysis of the game. He generously sent me the actual computations a few weeks later, consisting of two large boxes of lab manuals filled with thousands of pages of calculations done by the authors on desk calculators while they served in the army. During the spring of 1959, wedged in between my teaching duties and research in the UCLA Mathematics Department, I mastered every detail, my excitement mounting as I strove to speed up the enormous number of calculations that lay between me and a winning system.
The Baldwin strategy was the best way to play the game when nothing was known about which cards had already been played. Their analysis was for a single deck because that was the only version played in Nevada at the time. The Baldwin group also showed that the advice of the reigning gambling experts was poor, unnecessarily giving the casinos an extra 2 percent advantage.
Any strategy table for blackjack must tell the player how to act for each case that can arise from the ten possible values of the dealer’s upcard versus each of the fifty-five different pairs of cards that can be dealt to the player. To find the best way for the player to manage his cards in each of these 550 different situations, you need to calculate all the possible ways subsequent cards can be dealt and the payoffs that result. There may be thousands, even millions of ways each hand can play out. Do this for each of the 550 situations and the computations just for the complete deck become enormous. If you are dealt a pair, the strategy table must tell you whether or not to split it. The next decision is whether or not to double down, which is to double your bet and draw exactly one card to the first two cards of a hand. Your final decision is whether to draw more cards or to stop (“stand”). Once I had figured out a winning strategy, I planned to condense these myriad decisions onto tiny pictorial cards, just as I had with the Baldwin strategy. This would allow me to visualize patterns, making it much easier to recall what to do in each of the 550 possible cases.
The Baldwin group’s calculations for the full deck were approximate because the exact calculations could not be done with desk calculators in a human lifetime. The work I faced in 1959 was much more extensive as I had to deduce the strategy for each of the millions of possible partially played decks. To see what I was up against, suppose, as was standard practice then, the dealer begins by “burning” one card. This means taking it from the top of the deck and placing it on the bottom faceup as a signal not to deal it later, leaving fifty-one cards still in play. There are ten cases to analyze, corresponding to the ten different card values of the missing card: Ace, 2…9, 10. What if, as often happens, we see the burned card and want to use the knowledge that it’s gone? We could apply the Baldwin analysis for each of these ten cases and make a strategy table for each of 550 playing situations. We would then have eleven strategy tables, one for the full deck and one each for the ten possibilities where one card is missing.
Next, suppose we identify two missing cards, so only fifty cards are left to play. How many different such fifty-card decks can arise? As there are forty-five ways to take out two cards with different values [(A, 2), (A, 3)…(A, 10); (2, 3), (2, 4)…(2, 10); et cetera] and ten ways to take out two cards of the same value [(A, A), (2, 2)…(10, 10)], the total is fifty-five. This leads to fifty-five more calculations, and fifty-five more strategy tables, each of which could take twelve man-years if done on desk calculators as per the Baldwin group. We could continue in this way to develop strategy tables for every such partial deck. For one deck of fifty-two cards there are about thirty-three million of these partially played decks, leading us to a gigantic library of thirty-three million strategy tables.
Facing four hundred million man-years of calculations, with a resulting railroad car full of strategy tables, enough to fill a Rolodex five miles long, I tried to simplify the problem. I predicted that the strategies and player’s edge for partly used decks depended mainly on the fraction—or, equivalently, the percentage—of each type of card remaining rather than on how many there were.
This turned out to be true, and it meant, for example, that the effect of 12 Tens when, say, forty cards were left to be played, was about the same as 9 Tens with thirty cards left, and 6 Tens with twenty left, as all three of these decks have the same fraction, 3⁄10, or 30 percent, of Ten-value cards. In counting cards, it’s mainly the fraction remaining that matters, not the number.
I started by looking at how the player’s strategy and edge changed when I varied the percentage of each card. I planned to take out all four Aces, do the calculations, and see what happened, then repeat this by removing only four 2s, then only the four 3s, and so forth.
I began this work during the spring semester of 1959. I was teaching then at UCLA for the year after I received my PhD in June 1958. That happened because I had gotten my degree sooner than either I or my adviser, Angus Taylor, had expected. As a result I hadn’t applied for a postdoctoral teaching position, thinking it would be another year before I was available. Professor Taylor arranged for my interim appointment at UCLA and then helped me find possible positions for the following year. The offers I liked best were a C. L. E. Moore Instructorship at the Massachusetts Institute of Technology (MIT), and a job at the General Electric Corporation in Schenectady, New York. At GE I would be using my physics background to do orbital calculations for space projects. This sounded like it would be interesting for a while, but I didn’t think I would have the freedom I expected to find in academia to follow my interests wherever they led. Expecting that kind of life as a professor in a university, I chose MIT as a first step.
We moved to MIT in June 1959. To get there, I purchased a used black Pontiac sedan for $800 at a police auction and drove it across the country towing a two-wheeled U-Haul trailer loaded with our household goods. We were expecting our first child in two months, so Vivian stayed with her parents in Los Angeles while I went to Cambridge, Massachusetts, to set up our apartment and do mathematics research on a summer grant. As I was obliged under the terms of the grant to work at MIT until mid-August and the baby was due a few days later, I was very nervous about whether I would get back in time. Vivian and I spoke almost daily that summer over the phone. Fortunately, the results of her checkups were always excellent.
Two Japanese mathematicians who were visiting at UCLA needed a ride to New York. I was happy to take them, in return for their sharing the driving. But on a deserted highway somewhere in Ohio, I was startled from a sound sleep about 1 A.M. as the brakes squealed and the car shuddered. We stopped just feet from a large brown-and-white cow meandering placidly across the road. Since the only set of brakes we had were on the car, and the loaded trailer doubled our mass, it also doubled our stopping distance. I had explained this carefully before we left, but apparently without success. Fighting fatigue, I drove the rest of the way.
Once I reached Cambridge, I had a lot to think about. I had never been to the Boston area and didn’t know anyone there. Most of the regular staff and faculty were away for the summer but the department did arrange a marvelous rental, the first floor of a grand old three-story family home in Cambridge. Having taken it sight unseen, I was pleasantly surprised at how large it was and at the graciousness of my landlady, an Irish widow who lived there with the two youngest of her five sons.
By day I did academic research in mathematics, but after dinner I’d walk through the nearly deserted buildings to the calculator room. Once there, I would pound the Monroe calculators every night from eight o’clock until shortly before dawn. These were noisy electromechanical beasts about the size of a very large typewriter. They could add, subtract, multiply, and divide and were equivalent in this to today’s simplest cheapest handheld digital devices. As there was no air-conditioning, I worked shirtless, my fingers flying over the clacking keyboard, the calculator whirring and rumbling in the humid Cambridge summer nights.
One morning about three o’clock, I came out to find my car was missing from the spot where I regularly parked it. When I went back inside to call the police, a friendly night owl graduate student told me that the officers of the law might themselves be the problem. I phoned the police station and learned my car had been towed. When I pointed out that it had been legally parked, the officer on duty explained that since it was seen in the same spot every night, they assumed it was abandoned. I hurried downtown to night court where the judge to whom I appealed screamed and threatened to fine me $100 on the spot if I said another word. The friendly student, who had driven me there, explained that the police had an arrangement with the tow yard and that the impound charges for my car were going to go up quickly if I pressed my case. The next morning I ransomed my car at the tow yard for about $100. This was a week’s pay. Welcome to Boston. Fortunately my new hometown was also a beautiful one, rich in science, education, culture, and the arts.
The weeks wore on and the calculations piled up. However, even though I had introduced shortcuts and efficiencies and was very fast, I was making little progress. My hand calculations were going to take hundreds, perhaps thousands of years. At that point I learned that MIT had an IBM 704 computer and, being a faculty member, I could use it. Using a book from the computer center, I taught myself to program the machine in its language, FORTRAN.
In August 1959 I flew to Los Angeles four days before the birth of our first child. Knowing we were having a girl, we agonized over name choices for weeks, finding many we liked but none that was a first choice for both of us. We enlisted the help of Vivian’s brother, Ray, a speech major at UCLA with a gift for the English language, who would go on to a distinguished legal career. He invented the name Raun, with its uplifting rhyming images, like dawn and fawn. None of us had ever heard of it but we loved it and the search was over.
I was back at MIT a month later with Vivian and our new baby, beginning my teaching and research duties. MIT then, as now, had one of the best mathematics departments in the world, and much was expected of its young faculty members. I taught two classes each semester, which meant six hours per week in the classroom, preparation that could run another twelve to fifteen hours a week, additional hours in my office to meet with and help students, plus the giving and grading of homework and exams. We were also expected to conduct and publish our own original research in scholarly journals. When this was submitted, it was reviewed by anonymous experts, known as referees, as a precondition of acceptance. Rejections were common. Those of us who wanted to succeed in the academic hierarchy all knew the mantra “Publish or perish.” Despite all this, I also continued to work on my “arbitrary subsets” blackjack program for the IBM 704 computer, testing and correcting the computer code for one module (or “subroutine”) at a time.
The 704 was one of the early mainframe electronic computers, one of a series of increasingly powerful models developed by IBM. In those days, users entered instructions via punched cards roughly the size of a $1 bill. A card had eighty columns with ten oblong vertical marks in each column. I put cards, one at a time, in a keypunch and typed as I would on a typewriter; each time I hit a key the machine punched holes in a vertical line and shifted to the next column. The pattern of holes represented the letter, number, or symbol on that key.
I left batches of punched cards bound with a rubber band in the in-bin at the computer center, where they were collected and read as instructions to be carried out by the IBM 704. It took several days for me to get the results, because MIT shared the computer with thirty New England universities (such as Amherst, Boston College, and Brandeis).
The work went more quickly as I mastered the strange new language. I had divided the problem of writing the computer program into sections, or subroutines, each of which I tested, corrected, and then cross-checked. Weeks passed, then months, as I completed one part after another. Finally, early in 1960, I put them together and submitted the complete program. The first results indicated that the casino advantage, when you played as perfectly as possible without keeping track of the cards that have been played, was 0.21 percent. The game was virtually even for anyone. It wouldn’t take much in the way of card counting to give the player an edge! However, because even the IBM 704 was unable to do all the necessary calculations in the available time, I used approximations for some parts of the calculations. I knew that the results using these shortcuts were slightly pessimistic. This meant that the real game was even better for the player than my computer results indicated.
As computers became more powerful, my approximations were removed step by step. Twenty years later, around 1980, computers had finally become powerful enough to show that the final figure for one deck using the blackjack rules as given in the book I would go on to write, Beat the Dealer, was +0.13 percent in favor of the player. Players using my strategy had had, all along, a small edge over the casino even without keeping track of the cards. But the real power of my method was that I could analyze the game not only for the complete deck, but for any collection of cards. I could explore the impact on the game as the cards were used during play.
Now I instructed the computer to venture into the unknown: Analyze the game when all four Aces were missing. Comparing the results with those I already had for the full deck, I would see the effect Aces had on the game. With anticipation I picked up my rather thick deck of punch cards a few days later from the out-bin. (It occurred to me that I was using cards to evaluate a card game.) The IBM 704 had done a thousand man-years of hand calculations in just ten minutes of computer time. I looked at these results with great excitement, for they would very likely either prove I was right or dash my hopes. The result was a player disadvantage of 2.72 percent with all the Aces gone—2.51 percent worse than the overall 0.21 percent casino edge. Although this was a huge shift in favor of the casino, it was actually great news.
It proved conclusively what I believed in that Eureka moment back in the UCLA library when I thought I could beat the game, namely, that as cards were played there would be huge shifts in the edge, back and forth, between the casino and the players. The math also showed me that if removing a specific group of cards from the deck shifted the odds in one direction, adding an equal number of the same cards instead would move the odds the other way about the same amount. This meant that with an Ace-rich deck rather than an Ace-poor deck, the player should have a big advantage. For instance, with twice the usual proportion of Aces, which happens when all four Aces remain among the last twenty-six cards (half the deck), the player ought to gain roughly 2.51 percent over his initial 0.21 percent disadvantage, for a net edge of about 2.30 percent.
Every two or three days I went to the computer center and picked up another completed calculation, each of which would have taken a thousand man-years by hand. I now knew the impact of removing any four cards of one type from the deck. Taking out four Aces was worst for the player, and removing four Tens was next worst, adding 1.94 percent to the house edge. But taking out the “small” cards, which are 2, 3, 4, 5, and 6, helped the player enormously. Removing four 5s was best, changing the casino edge of 0.21 percent to a huge player edge of 3.29 percent.
Now I could design a great variety of winning strategies based on keeping track of the cards. My analysis using MIT’s IBM 704 had produced the basic results that gave me the Five-Count System, most of the Ten-Count System, and the ideas for what I called the ultimate strategy. The latter assigned a point value to every card, proportional to its effect on the game, with Aces each counted as −9, 2s counted as +5, and so on, down to Tens counted as −7. Though this was too difficult for almost anyone to keep track of mentally, many simpler counting systems worked quite well. One of the best compromises between ease of use and profitability was to count the small-value cards (2, 3, 4, 5, 6) as +1 as they are seen during play, intermediate cards (7, 8, 9) as 0, and large-value cards (10, J, Q, K, A) as −1. From the results of my computer runs anyone could work out the details of nearly all the blackjack card counting systems in use today.
Intuitively, these results make sense. For instance, when the dealer has a total of 16 he has to hit. He loses if he draws a big card that puts his total over 21 and survives if he draws a small card. A 5 gives him 21, best of all. So he benefits when the deck is richer in small cards and poorer in big cards. On the other hand, when the deck has a higher percentage of Aces and Tens, there will be more two-card totals of 21, or blackjacks. The player and the dealer each win with a blackjack about 4.5 percent of the time but the player gets paid 1.5 times his bet for this while the dealer gains only the player’s bet, for a net benefit to the player.
Keeping track of the 5s leads to a very simple winning system. Suppose the player bets small whenever any 5s are left and bets big whenever all the 5s are gone. The likelihood of all the 5s being gone increases as fewer cards remain. When twenty-six cards are left, this will happen about 5 percent of the time, and if only thirteen cards are left, 30 percent of the time. Since the player then has a 3.29 percent edge on his bets, if these are very big compared with his other bets he wins in the long run.
For actual casino play, I built a much more powerful winning strategy based on the fluctuation in the percentage of Ten-value cards in the deck, even though my calculations showed that the impact of a Ten was less than that of a 5, since there were four times as many Tens. The larger fluctuations in “Ten-richness” that resulted gave the player more and better opportunities.
During our family drive from Boston to California in the summer of 1960, I persuaded a reluctant Vivian to stop briefly in Las Vegas so I could test the Tens strategy. We sat down to play blackjack in one of the casinos downtown on Fremont Street. I had a $200 bankroll (worth about $1600 in 2016) and a palm-sized card with my new strategy. I hoped not to use the card and so avoid drawing attention to myself. This card was unlike anything before it. Not only did it tell me how to play every hand versus each dealer upcard, but it also showed how much to bet and how the playing decisions changed as the percentage of Tens varied. Specifically, the complete deck had 36 non-Tens and 16 Tens, so I started counting “36, 16,” which gives a ratio of non-Tens to Tens of 36 ÷ 16 = 2.25.
Vivian and I sat down together, with her betting 25 cents a hand to keep me company. As play progressed and I kept track of the non-Tens and Tens that were used, I reduced the totals for those remaining. Whenever I had to place a bet or make a decision on how to play my hand, I used the current totals to recalculate the ratio. A ratio below 2.25 meant the deck was Ten-rich, and when a ratio hit 2.0 the player had an edge of about 1 percent. For ratios of 2.0 or less, which meant advantages of 1 percent or more, I bet between $2 and $10 depending on the size of my edge. Otherwise I bet $1.
Vivian watched nervously as I gradually lost $32. At this point my dealer said hostilely, “You’d better take out some more money, because you’re going to need it.” Smelling a rat, Vivian said, “Let’s get out of here.” Even though I lost, I was satisfied because I had shown that I could play the Ten-Count System at casino speed without looking at the strategy card. The $32 loss was well within the range of possible outcomes predicted by my theory, so it didn’t lead me to doubt my results. With nothing more for me to learn that day, I left, poorer once again but, I hoped, wiser.
Mathematical friends at MIT were astounded that fall when I told them of my discovery. Some thought I should publish quickly to establish priority before someone else either rediscovered my idea or stole it and passed it off as their own. I needed little prompting, since I had already been burned once. While I was at UCLA, my PhD thesis adviser, Angus Taylor, suggested that I send some of my mathematical work to a well-known California mathematician for his comments. I got no response. But eleven months later at a Southern California meeting of the American Mathematical Society, Taylor and I heard the great man talk. The subject was my discovery, in detail, presented as part of his original work, and it was also about to appear under his name in print, in a well-known mathematical journal. Both of us were stunned. Taylor, who would later become academic vice president of the entire University of California system, was an ethical and experienced mathematician to whom I looked for guidance, but he didn’t know what to do. So neither of us did anything.
It is also common in science for the time to be right for a discovery, in which case it is made independently by two or more researchers at nearly the same time. Famous examples include calculus by Newton and Leibniz, and the theory of evolution by Darwin and Wallace. Five years before I did my blackjack work, it would have been much more difficult to accomplish. Five years afterward, with the increasing power and availability of computers, it was clearly going to be much easier.
Another reason to publish quickly is the well-known phenomenon that it is typically much easier to solve a problem if you know it can be solved. So the mere fact that the news was spreading through word of mouth meant others would repeat my work, sooner rather than later. This point was made in a science-fiction story I had read earlier in college. A professor at Cambridge University has by far the most brilliant class of graduate physics students ever. He divides the twenty of them into four teams of five and assigns his hardest homework problems. Since the class knows he has the answers, they persist until they can answer every question. Finally, to stump them, he says, untruthfully, that the Russians have discovered how to neutralize gravity, and their job is to show how it’s done. A week later two of the four groups present solutions.
To protect myself from this happening with my work on blackjack, I settled on Proceedings of the National Academy of Sciences, as it was the quickest to publish of any journal I knew, taking as little as two or three months, and was also very prestigious. This required a member of the academy to approve and forward my work, so I sought out the only mathematics member of the academy at MIT, Claude Shannon. Claude was famous for the creation of information theory, which is crucial for modern computing, communications, and much more.
The department secretary arranged a short appointment with a reluctant Shannon at noon. However, she warned me that Shannon was going to be in for only a few minutes, that I shouldn’t expect more, and that he didn’t spend time on topics or people that didn’t interest him. A little in awe but feeling lucky, I arrived at Shannon’s office to find a thinnish alert man of middle height and build, somewhat sharp-featured. I told the blackjack story briefly and showed him my proposed article.
Shannon cross-examined me in detail, both to understand the way I analyzed the game and to find possible flaws. My few minutes turned into an hour and a half of animated dialogue, during which we grabbed lunch in the MIT cafeteria. He pointed out in closing that I appeared to have made the big theoretical breakthrough on the subject, and that what remained to be discovered would be more in the way of details and elaboration. He asked me to change the title from “A Winning Strategy for Blackjack” to “A Favorable Strategy for Twenty-One,” as this title was more sedate and acceptable to the academy. Space in the magazine was tight, and each member could submit only a limited number of pages per year, so I reluctantly accepted Shannon’s suggestions for condensation. We agreed that I’d send him the revision right away to forward to the academy.
As we returned to the office he asked, “Are you working on anything else in the gambling area?” I hesitated for a moment then decided to spill my other big secret, explaining why roulette was predictable, and that I planned to build a small computer to make the predictions, wearing it hidden under my clothing. As I outlined my progress, ideas flew between us. Several hours later, as the Cambridge sky turned dusky, we finally parted, excited by our plans to work together to beat the game.
Meanwhile, I was planning to present my blackjack system at the annual meeting of the American Mathematical Society in Washington, DC. I submitted an abstract of my talk titled “Fortune’s Formula: The Game of Blackjack” for the program booklet (The Notices), where it would appear amid a large collection of typically technical and abstruse summaries of presentations.
When the screening committee received my abstract, their near-unanimous reaction was to reject it. I learned this later from John Selfridge, a number theorist whom I had known at UCLA and a member of the committee. For a while, he held the world’s record for finding the largest known prime number. (A prime is a positive whole number divisible only by itself and one. The first few are 2, 3, 5, 7, 11, 13…) Fortunately, Selfridge persuaded them that I was a legitimate mathematician and that if I said it was true, it likely was.
Why would the committee reject my talk? Professional mathematicians regularly receive claims that the sender has solved some famous problem, claims that almost always turn out to be from cranks, from the mathematically uneducated unaware of what’s already been done, or that include proofs containing simple errors. The so-called solution often is to a problem that has long ago been proven to be impossible, such as a method for trisecting (dividing into three equal angles) any angle whatsoever with compass and straightedge alone. On the other hand, students of plane geometry learn a simple method for bisecting an angle this way. A small change in the problem, from dividing an angle into two equal parts, to splitting it into three equal parts, transforms an easy problem into an impossible one.
The situation then was similar for gambling systems, since mathematicians had proved that a winning system was impossible for most of the standard gambling games. And obviously, if the casinos could be beaten they would either change the rules of the game or go out of business. No wonder the committee was inclined to reject my abstract. Ironically, their reason for doing so—that mathematicians had apparently proven that winning gambling systems were impossible—was my strongest motivation for showing it could be done.
Two evenings before I left for the meeting, I was surprised by a call from Dick Stewart of The Boston Globe, inquiring about my upcoming talk. Meanwhile, the newspaper sent out a photographer. I explained the basic ideas of my system on the phone. The next morning my picture and Stewart’s article were on the front page. Within hours the news services released the story and more photos to scores of papers across the country. As I left for the airport Vivian was wearily logging an incoming wave of hundreds of messages and, before long, our baby daughter, Raun, cried each time the telephone rang.