The Perfect Bet

by Adam Kucharski

  TOWARD THE END OF 1977, the New York Academy of Sciences hosted the first major conference on chaos theory. They invited a diverse mix of researchers, including James Yorke, the mathematician who first coined the term “chaotic” to describe ordered yet unpredictable phenomena like roulette and weather, and Robert May, an ecologist studying population dynamics at Princeton University.

  Another attendee was a young physicist from the University of California, Santa Cruz. For his PhD, Robert Shaw was studying the motion of running water. But that wasn’t the only project he was working on. Along with some fellow students, he’d also been developing a way to take on the casinos of Nevada. They called themselves the “Eudaemons”—a nod to the ancient Greek philosophical notion of happiness—and the group’s attempts to beat the house at roulette have since become part of gambling legend.

  The project started in late 1975 when Doyne Farmer and Norman Packard, two graduate students at UC Santa Cruz, bought a refurbished roulette wheel. The pair had spent the previous summer toying with betting systems for a variety of games before eventually settling on roulette. Despite Shannon’s warnings, Thorp had made a cryptic reference to roulette being beatable in one of his books; this throwaway comment, tucked away toward the end of the text, was enough to persuade Farmer and Packard that roulette was worth further study. Working at night in the university physics lab, they gradually unraveled the physics of a roulette spin. By taking measurements as the ball circled the wheel, they discovered they would be able to glean enough information to make profitable bets.

  One of the Eudaemons, Thomas Bass, later documented the group’s exploits in his book The Eudaemonic Pie. He described how, after honing their calculations, the group hid a computer inside a shoe and used it to predict the ball’s path in a number of casinos. But there was one piece of information Bass didn’t include: the equations underpinning the Eudaemons’ prediction method.

  MOST MATHEMATICIANS WITH AN interest in gambling will have heard the story of the Eudaemons. Some will also have wondered whether such prediction is feasible. When a new paper on roulette appeared in the journal Chaos in 2012, however, it revealed that someone had finally put the method to the test.

  Michael Small had first come across The Eudaemonic Pie while working for a South African investment bank. He wasn’t a gambler and didn’t like casinos. Still, he was curious about the shoe computer. For his PhD, he’d analyzed systems with nonlinear dynamics, a category that roulette fell very nicely into. Ten years passed, and Small moved to Asia to take a job at Hong Kong Polytechnic University. Along with Chi Kong Tse, a fellow researcher in the engineering department, Small decided that building a roulette computer could be a good project for undergraduates.

  It might seem strange that it took so long for researchers to publicly test such a well-known roulette strategy. However, it isn’t easy to get access to a roulette wheel. Casino games aren’t generally on university procurement lists, so there are limited opportunities to study roulette. Pearson relied on dodgy newspaper reports because he couldn’t persuade anyone to fund a trip to Monte Carlo, and without Shannon’s patronage, Thorp would have struggled to carry out his roulette experiments.

  The mathematical nuts and bolts of roulette have also hindered research into the problem. Not because the math behind roulette is too complex but because it’s too simple. Journal editors can be picky about the types of scientific papers they publish, and trying to beat roulette with basic physics isn’t a topic they usually go for. There has been the occasional article about roulette, such as the paper Thorp published that described his method. But though Thorp gave enough away to persuade readers—including the Eudaemons—that computer-based prediction could be successful, he omitted the details. The crucial calculations were notably absent.

  Once Small and Tse had convinced the university to buy a wheel, they got to work trying to reproduce the Eudaemons’ prediction method. They started by dividing the trajectory of the ball into three separate phases. When a croupier sets a roulette wheel in motion, the ball initially rotates around the upper rim while the center of the wheel spins in the opposite direction. During this time, two competing forces act on the ball: centripetal force keeping it on the rim, and gravity pulling it down toward the center of the wheel.

  FIGURE 1.1. The three stages of a roulette spin.

  The pair assumed that as the ball rolls, friction slows it down. Eventually, the ball’s angular momentum decreases so much that gravity becomes the dominant force. At this point, the ball moves into its second phase. It leaves the rim and rolls freely on the track between the rim and the deflectors. It moves closer to the center of the wheel until it hits one of the deflectors scattered around the circumference.

  Until this point, the ball’s trajectory can be calculated using textbook physics. But once it hits a deflector, it scatters, potentially landing in one of several pockets. From a betting point of view, the ball leaves a cozy predictable world and moves into a phase that is truly chaotic.

  Small and Tse could have used a statistical approach to deal with this uncertainty. However, for the sake of simplicity, they decided to define their prediction as the number the ball was next to when it hit a deflector. To predict the point at which the ball would clip one of the deflectors, Small and Tse needed six pieces of information: the position, velocity, and acceleration of the ball, and the same for the wheel. Fortunately, these six measurements could be reduced to three if they considered the trajectories from a different standpoint. To an onlooker watching a roulette table, the ball appears to move in one direction and the wheel in the other. But it is also possible to do the calculations from a “ball’s-eye view,” in which case it’s only necessary to measure how the ball moves relative to the wheel. Small and Tse did this by using a stopwatch to clock the times at which the ball passed a specific point.

  One afternoon, Small ran an initial series of experiments to test the method. Having written a computer program on his laptop to do the calculations, he set the ball spinning, taking the necessary measurements by hand, as the Eudaemons would have done. As the ball traveled around the rim a dozen or so times, he gathered enough information to make predictions about where it would land. He only had time to run the experiment twenty-two times before he had to leave the office. Out of these attempts, he predicted the correct number three times. Had he just been making random guesses, the probability he would have got at least this many right (the p value) was less than 2 percent. This persuaded him that the Eudaemons’ strategy worked. It seemed that roulette really could be beaten with physics.

  Having tested the method by hand, Small and Tse set up a high-speed camera to collect more precise measurements about the ball’s position. The camera took photos of the wheel at a rate of about ninety frames per second. This made it possible to explore what happened after the ball hit a deflector. With the help of two engineering students, Small and Tse spun the wheel seven hundred times, recording the difference between their prediction and the final outcome. Collecting this information together, they calculated the probability of the ball landing a specified distance away from the predicted pocket. For most of the pockets, this probability wasn’t particularly large or small; it was pretty much what they’d have expected if picking pockets at random. Some patterns did emerge, however. The ball landed in the predicted pocket far more often than it would have if the process were down to chance. Moreover, it rarely landed on the numbers that lay on the wheel directly before the predicted pocket. This made sense because the ball would have to bounce backward to get to these pockets.

  The camera showed what happened in the ideal situation—when there was very good information about the trajectory of the ball—but most gamblers would struggle to sneak a high-speed camera into a casino. Instead, they would have had to rely on measurements taken by hand. Small and Tse found this wasn’t such a disadvantage: they suggested that predictions made with a stopwatch could still provide gamblers with an expected profit of 18 percent.


  After announcing his results, Small received messages from gamblers who were using the method in real casinos. “One guy sent me detailed descriptions of his work,” he said, “including fabulous photos of a ‘clicker’ device made from a modified computer mouse strapped to his toe.” The work also came to the attention of Doyne Farmer. He was sailing in Florida when heard about Small and Tse’s paper. Farmer had kept his method under wraps for over thirty years because—much like Small—he disliked casinos. The trips he made to Nevada during his time with the Eudaemons were enough to convince him that gambling addicts were being exploited by the industry. If people wanted to use computers to beat roulette, he didn’t want to say anything that would hand the advantage back to the casinos. However, when Small and Tse’s paper was published, Farmer decided it was time to finally break his silence. Especially because there was an important difference between the Eudaemons’ approach and the one the Hong Kong researchers had suggested.

  Small and Tse had assumed that friction was the main force slowing the ball down, but Farmer disagreed. He’d found that air resistance—not friction—was the main reason for the ball slowing down. Indeed, Farmer pointed out that if we placed a roulette table in a room with no air (and hence no air resistance), the ball would spin around the table thousands of times before settling on a number.

  Like Small and Tse’s approach, Farmer’s method required that certain values be estimated while at the roulette table. During their casino trips, the Eudaemons had three things to pin down: the amount of air resistance, the velocity of the ball when it dropped off the rim of the wheel, and the rate at which the wheel was decelerating. One of the biggest challenges was estimating air resistance and drop velocity. Both influenced the prediction in a similar way: assuming a smaller resistance was much like having an increased velocity.

  It was also important to know what was happening around the roulette ball. External factors can have a big effect on a physical process. Take a game of billiards. If you have a perfectly smooth table, a shot will cause the balls to ricochet in a cobweb of collisions. To predict where the cue ball will go after a few seconds, you’d need to know precisely how it was struck. But if you want to make longer-term predictions, Farmer and his colleagues have pointed out it’s not enough to merely know about the shot. You also need to take into account forces such as gravity—and not just that of the earth. To predict exactly where the cue ball will travel after one minute, you have to include the gravitational pull of particles at the edge of the galaxy in your calculations.

  When making roulette predictions, obtaining correct information about the state of the table is crucial. Even a change in the weather can affect results. The Eudaemons found that if they calibrated their calculations when the weather was sunny in Santa Cruz, the arrival of fog would cause the ball to leave the track half a rotation earlier than they had expected. Other disruptions were closer to home. During one casino visit, Farmer had to abandon betting because an overweight man was resting against the table, tilting the wheel and messing up the predictions.

  The biggest hindrance for the group, though, was their technical equipment. They implemented the betting strategy by having one person record the spins and another place the bets, so as not to raise the suspicions of casino security. The idea was that a wireless signal would transmit messages telling the player with the chips which number to bet on. But the system often failed: the signal would disappear, taking the betting instructions with it. Although the group had a 20 percent edge over the casino in theory, these technical problems meant it was never converted into a grand fortune.

  As computers have improved, a handful of people have managed to come up with better roulette devices. Most rarely make it into the news, with the exception of the trio who won at the Ritz in 2004. On that occasion, newspapers were particularly quick to latch on to the story of a laser scanner. Yet when journalist Ben Beasley-Murray talked to industry insiders a few months after the incident, they dismissed suggestions that lasers were involved. Instead, it was likely the Ritz gamblers used mobile phones to time the spinning wheel. The basic method would have been similar to the one the Eudaemons used, but advances in technology meant it could be implemented much more effectively. According to ex-Eudaemon Norman Packard, the whole thing would have been pretty easy to set up.

  It was also perfectly legal. Although the Ritz group were accused of obtaining money by deception—a form of theft—they hadn’t actually tampered with the game. Nobody had interfered with the ball or switched chips. Nine months after the group’s initial arrest, police therefore closed the case and returned the £1.3 million haul. In many ways, the trio had the UK’s wonderfully archaic gambling laws to thank for their prize. The Gaming Act, which was signed in 1845, had not been updated to cope with the new methods available to gamblers.

  Unfortunately, the law does not hand an advantage only to gamblers. The unwritten agreement you have with a casino—pick the correct number and be rewarded with money—is not legally binding in the UK. You can’t take a casino to court if you win and it doesn’t pay up. And although casinos love gamblers with a losing system, they are less keen on those with winning strategies. Regardless of which strategy you use, you’ll have to escape house countermeasures. When Hibbs and Walford passed $5,000 in winnings by hunting for biased tables in Reno, the casino shuffled the roulette tables around to foil them. Even though the Eudaemons didn’t need to watch the table for long periods of time, they still had to beat a hasty retreat from casinos on occasion.

  AS WELL AS DRAWING the attention of casino security, successful roulette strategies have something else in common: all rely on the fact that casinos believe the wheels are unpredictable. When they aren’t, people who have watched the table for long enough can exploit the bias. When the wheel is perfect, and churns out numbers that are uniformly distributed, it can be vulnerable if gamblers collect enough information about the ball’s trajectory.

  The evolution of successful roulette strategies reflects how the science of chance has developed during the past century. Early efforts to beat roulette involved escaping Poincaré’s third level of ignorance, where nothing about the physical process is known. Pearson’s work on roulette was purely statistical, aiming to find patterns in data. Later attempts to profit from the game, including the exploits at the Ritz, took a different approach. These strategies tried to overcome Poincaré’s second level of ignorance and solve the problem of roulette’s outcome being incredibly sensitive to the initial state of the wheel and ball.

  For Poincaré, roulette was a way to illustrate his idea that simple physical processes could descend into what seems like randomness. This idea formed a crucial part of chaos theory, which emerged as a new academic field in the 1970s. During this period, roulette was always lurking in the background. In fact, many of the Eudaemons would go on to publish papers on chaotic systems. One of Robert Shaw’s projects demonstrated that the steady rhythm of droplets from a dripping tap turns into an unpredictable beat as the tap is unscrewed further. This was one of the first real-life examples of a “chaotic transition” whereby a process switches from a regular pattern to one that is as good as random. Interest in chaos theory and roulette does not appear to have dampened over the years. The topics can still capture the public imagination, as shown by the extensive media attention given to Small and Tse’s paper in 2012.

  Roulette might be a seductive intellectual challenge, but it isn’t the easiest—or most reliable—way to make money. To start with, there is the problem of casino table limits. The Eudaemons played for small stakes, which helped them keep a low profile but also put a cap on potential winnings. Playing at high-stakes tables might bring in more money, but it will also bring additional scrutiny from casino security. Then there are the legal issues. Roulette computers are banned in many countries, and even if they aren’t, casinos are understandably hostile toward anyone who uses one. This makes it tricky to earn good profits.

  For these reasons,
roulette is really only a small part of the scientific betting story. Since the shoe-computer exploits of the Eudaemons, gamblers have been busy tackling other games. Like roulette, many of these games have a long-standing reputation for being unbeatable. And like roulette, people are using scientific approaches to show just how wrong that reputation can be.

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  A BRUTE FORCE BUSINESS

  OF THE COLLEGES OF THE UNIVERSITY OF CAMBRIDGE, GONVILLE and Caius is the fourth oldest, the third richest, and the second biggest producer of Nobel Prize winners. It’s also one of the few colleges that serves three-course formal dinners every night, which means that most students end up well acquainted with the college’s neo-Gothic dining hall and its unique stained glass windows.