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Jess Marcum, Mathematical Genius, and the History of Card Counting

by Allan Schaffer, Ph.D.
(From Blackjack Forum Volume XXIV #3, Summer 2005)
© 2005 Blackjack Forum

[Editor’s Note: The thing that keeps striking me about Jess Marcum is the depth and intensity of respect and loyalty to him of the people who actually knew him and his achievements.

Marcum didn’t go for publicity or fame—he avoided it. He never published anything about his pioneering blackjack system. Yet 13 years after his death and over 50 years after his major accomplishments in card counting, he’s still vividly remembered by the professional gamblers who were active in Las Vegas then as the smart guy who figured it out, as well as to his later friends.

We are going to be documenting the history of those early days of card counting by publishing more articles on and by these early players in future issues of Blackjack Forum. –Arnold Snyder]

Introduction to Jess Marcum and the History of Card Counting

Jess Marcum, card counting pioneerJess Marcum developed a basic strategy for blackjack and was counting cards a decade before anyone else. Why is it then that not many of the members of today’s gambling community have heard of him? The reason is basically because he was a very private person. In addition, throughout his active gambling career he endeavored to keep his accomplishments secret to protect the opportunities he had discovered.

I am one of the few people who know much of the story. We were close friends for the last 15 years of his life and were also coworkers and partners for extended periods during that time. I had a number of opportunities to ask him questions about his adventures in casino gambling and his answers are the basis for much of this article.

My reasons for writing the article are both to augment the history of blackjack and to correct or verify, as the case may be, some of the things that have been written about Jess.

Jess Marcum at Rand Corporation

Jess Ira Marcum (originally Marcovitch) was born in Knoxville, Tennessee on December 30, 1919. His father had emigrated from Russia in 1904 at the age of 14 and was an entomologist and professor at the University of Tennessee. His mother, a librarian, had come from Austria in 1902. Both were fluent in Yiddish.

There is little evidence in what is known about his early life to suggest that he was destined to become a genius in mathematical analysis. He graduated from college as an electrical engineer about 1940 and went to work for the Westinghouse Corporation in Pittsburgh. His duties later took him to England. After the war he studied for a Ph.D., I believe, at Princeton University, but quit because he lost patience with formal education (perhaps a sign of genius).

In 1947 he joined the highly respected Rand Corporation think-tank located at Santa Monica, California. He was not a founder of Rand, as reported in some articles, but by solving extremely difficult problems that no one else could, he became Rand’s preeminent mathematician with an international reputation. This was an astonishing accomplishment in view of the fact he had only a BS degree and Rand was loaded with brainpower and Ph.D. mathematicians. His genius had emerged.

His principal interest at Rand was statistical physics with emphasis on nuclear radiation propagation, nuclear effects and the processing of electromagnetic wave data. His premiere accomplishment was formulating the original equations for calculating the radar backscatter from a steady target in a noisy background, the so-called signal-to-noise ratio on which all radar design is based. Since the government classified this work as secret, it was many years before the scientific literature could acknowledge it as the Marcum equation.

Thus, his two premiere mathematical accomplishments in life—supplying the formulation for radar design and beating blackjack with card counting—were not widely known. However, this lack of (deserved) fame did not bother Jess at all. He was content with solving problems that others could not and beating the odds.

Marcum Leaves Rand for Gambling

Jess became enchanted with the mathematical analysis of casino gambling around 1949. A few years later he left California for the casino world and lived in that environment for the next 15 to 20 years. Although he still consulted on scientific matters, gambling activities consumed most of his time. Later in this article I will relate what he told me about his blackjack adventures during this period.

Sometime in the late 1960’s or early 1970’s Jess moved back to California and again made scientific work his principal activity. However, he continued to be in high demand as a consultant to both the gambling industry and to gaming commissions. For his scientific habitat in California he joined a company named RDA, formed mostly by ex-Rand luminaries. It was there that I met Jess. We worked together at RDA for a number of years and starting in 1977 were partners in a gambling-related sideline adventure that lasted for almost a decade.

In the late 1980’s Jess became addicted to the sleeping pill Halcion and it almost killed him. Some of the side effects were deep depression, difficulty in concentrating, and physical deterioration. Relatives and friends, myself included, tried unsuccessfully to convince him to seek medical help. Finally one friend literally forced him to go to a hospital and, once there, he made a remarkably fast and almost complete recovery.

He then moved to Reno, his favorite place, where he spent the last few years of his life living in a hotel. In 1990 he briefly returned to casino consulting in an episode so spectacular that it made the front page of the Wall Street Journal. His efforts rescued the Trump Plaza Casino from a mysterious “whale” who had, in a previous visit, won six million dollars at baccarat. I will give some more information about this episode near the end of the present article.

Jess died from natural causes in Reno at the age of 72. He had been dead for several days when his body was discovered in his room by a hotel employee. His death certificate is dated January 16, 1992.

Jess Marcum Invents Card Counting

What I know about Jess and blackjack comes directly from his answers to my questions. It was not his manner to volunteer information about his exploits, but I was curious and he answered each of my queries straightforwardly. In retrospect I wish that I had asked more questions.

You may be wondering why he gave me this information when, to my knowledge, he never provided it to anyone else. The reason is probably that our close relationship came late in his career when his casino playing days were long over.

One day in 1949 a coworker at Rand, who often visited Las Vegas on weekends to gamble, talked Jess into accompanying him on one of these trips. At that time Jess knew little about casino gambling. He spent this first visit to Las Vegas observing the various games, particularly blackjack, which at that time was played with a single deck dealt down to the last card.

On the drive back to California he shared with his friend the (then startling) observation that, if one were to keep track of the cards previously dealt from the blackjack deck, one would have a significant advantage at the end. In fact, one could know exactly what the last card was going to be. He said that he was going to analyze this situation and felt obliged to offer his friend the opportunity to join him. His friend declined. I believe that this was the only time that Jess ever considered partnering with anyone in blackjack.

Jess carried out his analysis of blackjack with only pencil, paper and his mathematical genius. Jess never used a computer to solve a problem; in fact he was, by choice, computer illiterate. He solved everything analytically; that is, he derived equations to obtain his answers. In some cases, once he had derived the relevant equations, he might have an associate carry out any ensuing routine calculations on a computational machine of some type, but in the case of blackjack, I do not believe that even this secondary association with such machines occurred.

The equations that he developed for blackjack produced both a strategy for betting and algorithms for recording the running card count and translating the results into an actionable assessment. To my knowledge his notes have not survived, but it is safe to say that his feat has never been duplicated, since subsequent blackjack analyses have largely been based on computer simulations whereby many thousands of hands are played to analyze each situation statistically.

Jess determined that his overall advantage against the casinos was about 3%, a number within the range of later results quoted for computer simulations with similar systems.

When Jess moved full force on the Las Vegas casinos in the early 1950’s, they literally never knew what hit them. He was aware that counting cards was tedious and that realizing the fruits of a 3% advantage required many long and patient hours, but he also knew that the outcome was a certainty.

He won steadily. He tried to be as inconspicuous as he could, but before too long he began to attract the attention of casino management. The pit bosses hovered over him continuously, but, although it was evident to them that he had a never-before-seen winning method, they had no idea as to how he was accomplishing this feat.

Nine months passed and they were still baffled. Finally in desperation the Las Vegas casinos joined together and banned him from further play. He might be the first person ever banned anywhere for simply being too good at gambling. No one knows how much money he won during this, or any of his subsequent, blackjack ventures.

After Las Vegas the next target area was Reno. The story was the same. He won steadily and the casinos were completely befuddled. After six months, desperate, they engaged a detective, who shortly after uncovered Jess’s Las Vegas exploits. Of course the Reno casinos immediately banned him.

He then moved on to various other casino locations. One place that he mentioned was Hot Springs, Arkansas, at that time a gambling hotbed. Another was Havana. Although I never personally discussed with him any details of his Havana venture, Sam Cohen (whose biography of Jess I will deal with later) reports seeing a clipping from the front page of the Miami Herald that described Jess’s triumph in Havana.

Jess Marcum and Ed Thorp

I was surprised one day when Jess made a less than complimentary remark about the blackjack analyses of Dr. Edward Thorp, who wrote the classic book Beat the Dealer published in 1962. The problem, as Jess expressed it to me, was that Thorp had to use computer simulations to solve a problem that Jess had solved analytically a decade earlier. ⁱ

This criticism has always seemed unfair to me in view of the fact that Jess had chosen to keep his own work secret. In any event Jess said that he had no quarrel with the correctness of Thorp’s results and that they were consistent with his own.

Jess also told me that he had tried unsuccessfully via an emissary to dissuade Thorp from publishing the bookⁱⁱ. When I asked Jess why he cared, since I was under the impression that he had exploited all of the available opportunities, he growled back “there was still the Caribbean”. He meant that he had not as yet visited the Caribbean casinos, and in fact he never did. Apparently, however, he had covered all of the casinos in the U.S. as well as Havana and perhaps elsewhere.

Upon the publication of Thorp’s book, Jess chose to end his own active blackjack career. Beating the casinos had been his private province for about a decade, and he was no longer interested when the winning techniques became available to the public.

Although Dr. Thorp did not personally know Jess (out of courtesy he referred to him only as “the little dark-haired guy from Southern California”), he does include in his book a few of the stories about Jess that have circulated around the gambling world. Only one of these stories is, I believe, factual; namely that Jess was the sole player to have been barred from blackjack in Las Vegas up until that time.

Marcum and Ruchman

Peter Ruchman, in his year 2000 column entitled “How BJ Card Counting Really Started” and his follow-up 2001 column entitled “Thorp Steps Up to the Plate,” purports, among other things, to recount Jess’s blackjack adventures. Although Ruchman does properly deduce that Jess was the first person to calculate a Basic Strategy, his deduction is based on the idea that Jess was able to use a high powered computer for this purpose.

Ruchman says that Jess fed thousands of punch cards through an IBM 704 computer to develop his blackjack strategy, which he subsequently shared with a small group of Las Vegas gambling friends to form a winning consortium. What Ruchman wrote is very different from the account that Jess gave me. In particular, Jess took pride in the fact that he had developed his system mathematically.

Jess Marcum and Sam Cohen

Sam Cohen, who was a coworker and sometime friend of Jess’s, wrote Jess’s only biography, which he titled The Automat—Jess Marcum, Gambling Genius of the Century. One reason for my writing the present article is to comment on Cohen’s book.

It is unfortunate that his biography was written by a man who felt extremely hostile to Jess. The book contains many interesting facts and anecdotes about Jess. However, Cohen’s statements that Jess became sadistic, psychotic, and ultimately committed suicide are completely at odds with my experience and with the testimony of all of his friends and associates with whom I conferred. Jess did have some quirks, such as being exceedingly tight with money, but nothing that any of us considered highly abnormal.

I can think of two reasons why Cohen came to hate him. One is that Jess needled Cohen incessantly. The incentive for this needling probably stemmed from Jess’s intense interest in psychiatry, which rivaled his interest in mathematics. His psychiatric self training led him to analyze each of his acquaintances and then to treat them as he concluded that they deserved to be treated. He evidently concluded that Cohen deserved to be needled, but I know of no other person, regardless of status, that Jess treated uncongenially.

The other reason for Cohen’s animosity probably resulted from the strong motherly affection that Cohen’s wife displayed for Jess. She continually looked after Jess despite Cohen’s objections. When Jess went through his disastrous times with Halcion, Cohen’s wife and I, unknown to each other, were both with him a lot and concluded independently that his problems, which were severe, were almost entirely due to the pills. By Jess’s own statement to me later, 90% of his problems disappeared when he finally ended his addiction to the pills.

From the gambling viewpoint there are several apparent fallacies in Cohen’s book. He states that it “is meant to be a tale mainly about the life and exploits of Jess in the world of gambling”. However, Cohen’s account differs greatly from what Jess told me.

Moreover, as I will explain, some of what Cohen thought he knew appears to be based on disinformation. Cohen states that Jess’s original gambling accomplishment was the development of a system that won at horse racing, and that he moved to Las Vegas to exploit this system, was successful and “became the first horse better in Las Vegas history to be barred from betting”.

In actuality, Jess was not an expert in horse racing, nor was it ever really a major focus of his interest. However he did, while at Rand, develop two short-lived schemes that made him some money at horse racing.

One scheme involved using the suggested odds from a newspaper handicapper, which Jess determined could show a profit. This endeavor was terminated after a year or two by the death of the newspaper handicapper.

The other scheme involved win/show anomalies. By peering into the totelizator control room, Jess would surreptitiously observe the win and show pools at certain tracks where, at that time, this information was not displayed publicly on the tote boards. Whenever he detected a gross disparity in the wagering pools, he would make sizeable show bets.

Jess apparently used these two modest ventures in horse racing to mislead Cohen and others at Rand into believing that he was off to Vegas to play the races. Of course, he was going there to play his never-before-seen system for beating blackjack.

Cohen devotes just one page of his book to Jess’s blackjack adventures. He briefly summarizes what he had gleaned, or surmised, about Jess’s various exploits and states that Jess’s major forays were bankrolled by some big time gambling friends. This statement rings untrue to me. Jess was never in need of money and definitely not the type to share his gambling knowledge or winnings with anyone. I think that Cohen was again the victim of disinformation—this time to shield information about the magnitude of Jess’s winnings from others.

Cohen also says that while in Las Vegas Jess became a magnificent poker player. This statement too does not agree with what Jess told me. He said that, although he did play some in the big games in Las Vegas, he soon concluded that he was inherently not good enough to compete with the very top players and therefore lost interest in the game. [Editor’s note: This was a period when cheating was widespread in the poker rooms.]

Poker did not fall within Jess’s characterization of a potentially playable gamble because he could not definitively calculate the outcome. He gambled seriously only when he could calculate a positive return.

Jess Marcum’s Final Triumph

Although the vast majority of Jess Marcum’s gambling income came from playing blackjack, his gambling studies were certainly not limited to blackjack; in fact, he analyzed almost all of the common gambling games and devices from sports books to slot machines. In the various sports books he capitalized on occasional miscalculations by the oddsmakers. With regard to slot machines, his expertise in their statistics and fluctuations led the Nevada Gaming Commission to call on him often as a consultant.

It was principally in this role as consultant to the industry that Jess continued his association with gambling long after his active blackjack days were over. Very late in his life he gained national notoriety from a consulting assignment that he undertook for Donald Trump and his associates in Atlantic City, as I will discuss briefly next.

On June 28, 1990 Jess called me from his hotel in Reno and mentioned that there was an article concerning him on that day’s front page of The Wall Street Journal. It was entitled “Tale of a Whale: Mysterious Gambler Wins, Loses Millions” and it describes how Jess saved the Trump Plaza Casino from a mysterious gambler who, in his previous visit there, had won $6 million at baccarat.

This exciting story (it even includes a murder) was later masterfully researched and recounted by David Johnston in his book Temples of Chance, chapters 1 and 24. Johnston provides a realistic insight into the working of Jess’s mind as Jess combines his mathematical prowess with his (self-taught) psychiatric skill to harpoon the Whale.

In view of these extensive previous accounts of this episode, I am only going to summarize briefly Jess’s part in it. He described the whole thing rather succinctly to me. He said that in winning the first $6 million, the Whale had simply not played enough hands (700) to have a high probability of losing. Therefore, Jess advised the Trump people to just “let him play.” Although they were very nervous about following this advice, they did and ultimately the Whale lost $9.4 million.

I learned from the written accounts cited above that there were two key components to Jess’s analysis of this problem. First, he calculated that if the Whale could be enticed into playing 5000 hands or so, he would have only about a 15% chance of winning. However, since the Whale made a practice of quitting suddenly if he was winning significantly, a scheme was needed to ensure that he would not quit early.

Based on observing the Whale’s mannerisms at the gaming table, Jess concluded that the Whale would readily agree to a double-or-nothing rule; that is, play would end only when the gambler had either doubled his bankroll or lost it all. Jess calculated that the Whale was five times as likely to lose his bankroll as he was to double it.

As Jess (via his “psychoanalysis”) had predicted, the Whale agreed to the double-or-nothing rule. Play proceeded and, of course, turned out precisely as Jess had predicted. After 5,056 hands, the Whale had lost $9.4 million (84% of his bankroll). At that point, he quit in a huff.

Finale

After his Atlantic City triumph Jess returned to Reno and contented himself with exploiting sportsbook betting anomalies that sometimes arose due to calculational inaccuracies by the oddsmakers. He died suddenly less than two years after his return to Reno.

His epitaph should read mathematical genius and gambling legend. The scientific literature abundantly attests to the former. My hope is that this article will cement the latter. ♠

i Jess may possibly have been unaware of the early analytical work performed by Dr. Thorp prior to his resorting to the computer.

ii Dr. Thorp has informed me that he has no recollection of being contacted by an emissary from Jess Marcum.

Blackjack Insurance on Good Hands May Be A Good Idea After All

by Peter A. Griffin
(From Blackjack Forum Volume VIII #2, December 1988)
© Blackjack Forum 1988

[Note from Arnold Snyder: In the December issue of Blackjack Forum (Vol. VII #4), Marvin L. Master conjectured that if your card counting system indicated that the insurance bet was dead even, it may be advisable to insure a “good” hand, since this play would tend to reduce fluctuation. Marvin’s logic is clear. If the dealer does have a blackjack, then you will lose a bet you expected to win. Taking insurance would save this bet on one third of these hands, and on those hands where the insurance bet loses, you still expect to win your initial “good” hand. Thus, bankroll fluctuations are reduced.

Here now, to lay this controversy to rest, is Peter Griffin’s final word on whether and when you should take insurance on “good” blackjack hands. More probably, this article will give nightmares to players who consider attempting to work out Griffin’s insurance formula when playing.

Griffin shows that it is sometimes advisable to insure good hands—in order to reduce fluctuations—even when the insurance bet has a negative expectation! Unfortunately, most dealers only allow a couple of seconds for the insurance decision. So, the simplest answer is: Marvin was right! Insure your good hands when it’s a dead even bet.]

Marvin L. Master asks the question: Should you, to reduce fluctuations, insure a good hand when precisely one third of the unplayed cards are tens?

The answer depends upon what criterion for “reducing fluctuations” has been adopted. Griffin, in his monumental epic The Theory of Blackjack, shows that there are occasions when a Kelly proportional bettor would insure a natural with less than one third of the unplayed cards being tens.

Theoretically, this criterion could also be used to analyze whether to insure 20 and other favorable holdings. However, the answer is dependent upon both the fraction of capital bet and the distribution of the non-tens remaining in the deck.

An approximate calculation based upon what would seem a reasonable assumption in this regard suggested that 20 should be insured, but 19 not. Precise probabilities for the dealer were not computed, and the answer could well change if they were, or if a different fraction than assumed were wagered.

Another, more tractable, principle to reduce fluctuations also appears in The Theory of Blackjack: When confronted with two courses of action with identical expectations (the insurance bet here is hypothesized to neither increase nor decrease expectation), prefer that one which reduces the variance, hence average square, of the result.

This proves particularly easy to apply here. Let W, L and T stand for the probabilities of winning, losing, and tying the hand assuming insurance is not taken. In this case the average squared result is

ENx2 = 1 – T

If insurance is taken the average square becomes

EIx2 = 1/3 02 + W(1/2)2 + T(-1/2)2 + (L-1/3)(-3/2)2 = (W + T + 9L – 3)/4

Insurance will have a smaller average square if

W + T + 9L – 3 < 4 – 4T

Equivalently

W + 5T + 9L < 7

Or, subtracting

5(W + T + L) = 54L – 4W < 2L – W < .5L < W + .5

This will clearly be the case for player totals of 20, 19, 18, 11, 10, 9 and 8 if the dealer stands on soft 17. If the dealer hits soft 17, 18 would probably still be insurable, but not 8.

Returning to the Kelly criterion, the interested reader would be well advised to consult Joel Friedman’s “Risk-Averse” card counting and basic strategy modifications. Among Joel’s astute observations is that if a player confronts an absolute pick ’em hit-stand decision he should hit rather than stand. The reason is that he thereby trades an equal number of wins, (+1)2, and losses, (-1)2, for pushes, (0)2, thus reducing fluctuation. ♠

Can You Disguise Your Insurance Bet?

by Jake Smallwood
[From Blackjack Forum Vol. VIII #2, June 1988]
© 1988 Blackjack Forum

After a recent blackjack trip I was chastised by my teammates. They objected to the fact that, as a part of my card counting camouflage, I was taking insurance on everything but my minimum bet. The consensus was that I had fallen into “idiot” camouflage. I decided to get an idea of what my insurance yield should be and just how much I was giving up with my insurance camouflage plays. Most of my card-counting play (that particular trip) was on double deck blackjack games using the Hi-Lo count with a 1 to 20 spread ($10 to two hands of $100 actually). My betting scale is proportional only in the sense that my top bet is about half Kelly in size after accounting for two-hand covariance.

In order to simplify, I decided to set up three schedules of count per deck (true count) with the assumption that one deck of the two had been played. One schedule was developed with the quantity of each card type (plus, minus, zero valued cards) held constant for each table.

Of course, the sample would have been much more representative if I had prepared tables that took each group of cards with normal density, half normal density, and one-and-a-half normal. But that would be a lot of work and would require weighting by the probability of occurrence of each of the nine subsets. After all, I only wanted an indication, not a high-precision answer.

For all tables I assumed that 20% of the minus-valued cards (tens and aces) were aces. The result was a “profile” of a 52-card remainder for the counts between plus and minus six. I had some old frequency distribution approximations around which I used to weight for the occurrence of each count in the table. The insurance result was tabulated for each of the 52 cards in the remainder to estimate the edge at that count for that type of remainder subset.

The traditional blackjack player who insures at +3 or more count per deck would make about a 2.5% profit on insurance action. With a 20 to 1 spread the “insure all” approach would about break even.

The way I have been playing costs me about 15% of my potential insurance edge.

Actually, the cost is less since at +2 I am betting at two greens but only taking two reds of insurance. The pit knows I can’t be bothered taking more insurance with only two green out. I assume also that there is some element of risk aversion in the sense of fluctuation leveling with taking some negative expectation insurance. Any such effect is ignored.

Now, with my dumb style of play, I often get the pit personnel urging me to bet at 40 to 1 ratios and more. Well, what do you think? Based on the evidence, should I give up my idiot insurance camouflage?
[Snyder comments: It’s easy to see what Jake has done here and why his idiot camouflage works so well. Since he never insures his minimum bets, he doesn’t take insurance at those times when the house has its largest insurance advantage-at low counts. Since he always insures all bets above minimum, he always takes insurance when he has the advantage on these hands (high counts). The hands on which he took the insurance bet when the count was too low for this to be the proper play would for the most part be close to the “borderline” for taking insurance. Very little is actually given up on these incorrect insurance bets.

As a card counting camouflage strategy, Jake’s psychology is sound. Many gamblers and high rollers ignore insurance on insignificant bets. They seem to consider it a waste of time to insure piddling amounts of money.]
Insurance Return for Three Styles

Insure All Approach

Insure +3 or More (Traditional)

Insure All Except Minimum (My Style)

Notes

1. Sum count per deck of 3 to 6+.
2. Cumulative win for count per deck of 3 or more.
3. Sum count per deck 2 or more.
4. Cumulative win for count per deck of 2 or more. ♠

Will Chaos Theory Beat Keno?

by Arnold Snyder

(From Casino Player, August 1994)
© 1994 Arnold Snyder

Question:  According to a recent New York Times article, a computer whiz keno player in the new Montreal (Quebec) Casino used “chaos theory” to win $600,000 at keno one night in April. According to the article, the player picked 19 out of 20 keno numbers three times in a row! Then, the casino manager shut down the keno game.

Chaos theory, as I’m sure you are aware, is a relatively new branch of mathematics that attempts to find patterns in seemingly random results. If it’s true that a mathematician has now cracked this problem to the extent that what was believed to be random is now highly predictable, how will the casinos survive? Won’t new types of systems be developed, not only for keno, but also for games like roulette and craps and many others?

In fact, since blackjack is one of the few table games, along with baccarat, which — as card games — are dependent on less than perfectly random human shuffling, will we find that the truly random games, those which typically have a fixed house percentage, will become the new games of choice for professional players?

I’m not a mathematician, but I don’t understand how a game with a fixed house edge can be beaten in the long run. Isn’t this contrary to the generally accepted logic of mathematics? Please give us laymen the whole scoop on this new chaos theory, which must be the gambling system of the century, and may spell doom for the casinos as we know them.

How a Canadian College Student Beat Keno

Answer:  When this story broke, it was, as you might have guessed, of enormous interest to mathematicians. It seemed inconceivable that this Canadian college student, Daniel Corriveau, could have employed chaos theory — as he reportedly claimed he had done — to essentially predict the future in a game like keno. In fact, had Corriveau done what the newspapers reported he had done, his mathematical discovery would have such far reaching impact on virtually every branch of mathematics and science that the changes to the casino industry would pale in comparison to the advances you would see in electronics, chemistry, aviation, communications, etc., etc.

Unfortunately, when the whole story came out, it turned out that there wasn’t any great discovery with regards to applied chaos theory. There was simply an inadequate electronic number generating system in use at the Montreal Casino. Unlike the swirling ping pong ball systems used in many Nevada casinos, the Montreal Casino was using a state of the art computer random number generator to pick keno numbers. Corriveau simply discovered that the random number generator (RNG) was anything but random, and the only chaos theory he proved was that if you pick 19 out of 20 keno numbers three times in succession, casino management is thrown into chaos.

As RNGs are used these days in virtually all electronic slot machines, and are also frequently used by casino game analysts in simulating betting systems and player/house expectations, etc., allow me to explain what an RNG is, what it does, what it doesn’t do, how it works, and what went wrong in Montreal.

First of all, an RNG is not a piece of hardware, as you might normally assume from the term “generator.” It is a computer program, or more specifically, a mathematical formula, or a series of formulas, utilized to derive a “random” series of numbers. Can any mathematical formula actually produce a series of totally random numbers? No. And no RNG is truly random.

As an example, one RNG with which I am familiar is that used by Dr. John Gwynn in testing blackjack systems. His RNG has a cycle length of 2.7 billion. This means that his RNG will produce a string of 2.7 billion numbers which exhibit no discernible or predictable pattern, random for all intents and purposes, but that the cycle will begin repeating itself after 2.7 billion numbers.

All such RNGs require a “seed” number, which must be fed into the formula in order to start the series. With the RNG utilized by Dr. Gwynn, starting with a different “seed” does not change the 2.7 billion number cycle, but it will start the cycle at a different point. The same 2.7 billion number series, however, will be repeated.

In order to assure the most “random” results, many RNGs pick a seed number by utilizing whatever nanosecond the computer’s internal clock has at the moment the operator starts the program.

Such a random number generator works well for testing blackjack systems if what you want is “randomly” ordered cards, and your tests are going to be in the millions of hands, or even hundreds of millions of hands. If, however, Dr. Gwynn were attempting to pinpoint some expectation to umpteen decimal places, requiring a test of ten billion randomly dealt hands, his RNG would be inadequate because every 2.7 billion hands he would be recycling through the exact same series of cards.

How to Beat Keno: Find a Casino Without a Clock Chip

The problem in Montreal was that they did not purchase the clock chip for picking different seed numbers. As many Nevada casinos also use this same electronic keno number generator, and use it without the clock chip that generates seed numbers, Montreal management may have been under the impression that the seed generator was not a necessary component of the system, or that it was automatically included.

The major difference between Nevada casinos and the Montreal Casino, however, is that the Nevada casinos operate 24 hours per day, never turning off their keno games, while the Montreal Casino shuts down each night and reopens again in the morning. Without the clock chip to generate different seeds, each day the Montreal Casino was cycling through the same numbers, beginning at the same starting point! This is what Daniel Corriveau discovered. And this discovery paid him $600,000 in keno winnings.

So, unfortunately for mathematicians, there is no great breakthrough in chaos theory. Casinos can breathe a little easier, knowing that computer nerds will not start walking away with fortunes from games like keno and craps.

I will assume, of course, that those Nevada casinos which have not been using the clock chip on their electronic keno games, by now have purchased the chip, as this whole scenario has provided scam experts with one of the easiest, potentially most lucrative, and hitherto unknown “inside jobs” that could be imagined. In order to take a fortune from any casino in Nevada which uses this electronic keno random number generator, and which fails to utilize the clock chip for seed number generating, all a casino employee would have to do is turn the game’s power off for a moment at the end of each day, and the same number sequences would start repeating, just as they did in Montreal.

Incidentally, Daniel Corriveau was paid his $600,000 after investigators determined that he did not work in collusion with any casino employees. He took advantage of the game exactly as he found it. Hats off to Mr. Corriveau for teaching the Montreal Casino a lesson in mathematics. If there is a way to beat a game, someone will find it.  ♠

The First Unbalanced Point Count System

by Arnold Snyder
© 1983, 2005 Arnold Snyder

Blackjack Basic Strategy

Before you learn about counting cards, learn blackjack basic strategy. Even professional blackjack players use basic strategy to play most of their hands.

The Red Seven Count: Easy and Powerful

If you already know blackjack basic strategy, it’s time to learn an easy and powerful card counting system.

The easy Red Seven Count gets 80% of the potential gain available from the Hi-Lo Count and other counts that are significantly more difficult to learn and use. It is the strongest professional-level card counting system ever devised for its level of simplicity and ease of use.

The Red Seven Count

Blackjack players count cards to keep track of the proportions of high cards (Tens and Aces, the cards that are good for the player) and low cards (the cards that are good for the house) remaining in the decks to be dealt.

We don’t need to maintain separate counts of the Tens, fives, Aces, deuces, or any individual cards. We just need to know if the remaining deck has more high cards than normal or more low cards than normal.

In the Red 7 count, the high cards (Aces and Tens) are assigned the value -1, because each time one is dealt the remaining decks are a little poorer in the cards that are good for us.

Low cards (2, 3, 4, 5, and 6) are assigned the value +1, because each time one of these is dealt the remaining decks are a little better for us.

As for 8s and 9s, they are neutral cards, assigned a value of 0. This means that when we see them we ignore them, and don’t count them at all.

We count red 7s as +1, treating them like another low card. But we count black 7s as 0—that is, we ignore them as a neutral card. This is the device that creates the exact imbalance necessary for this count to work as an easy running count system, with no math at the tables. (Technically, it does not make any difference whether the red sevens or the black sevens are counted, so long as this precise imbalance is attained.)

Start learning to count cards by memorizing these values for each card denomination. Then practice keeping a running count by adding and subtracting these values from a starting count of 0 as you deal cards onto a table, one at a time from a deck. If the first card you turn over is a Jack, add -1 to your starting count so that your running count is -1. If the next card is a 6, add +1 to your count so that your running count is 0. If the next card is a 4, add +1 again so that your running count is +1.

Example:

Cards seen: 2, 6, A, 8, 9, X, X, 5

Point values: +1, +1, -1, 0, 0, -1, -1, +1

Running Count: +1, +2, +1, +1, +1, 0, -1, 0

By the time you get to the end of a single full deck of cards, your running count should be +2. If you have miscounted, try again. Then shuffle and go through the deck once more. Build up speed and accuracy, but do it at your own pace. Note that the deck ends at a running count of +2 because of those two extra red sevens we count as +1. They give the full deck 22 plus counts, against only 20 minus counts.

How to Practice Counting Cards

Practice, practice, practice.

Then learn to count as card counters actually do at the table—counting cards in groups.

When you are proficient at counting down a deck of cards one card at a time, practice turning the cards over two at a time, and count the cards in pairs. This is how you will do it at the casino tables, because it’s faster and easier for most people to count cards in pairs. This is because the cards in many pairs cancel each other out, so you don’t have to count them at all.

For example, every time you see a ten or an ace (both -1) paired with a 2, 3, 4, 5, 6, or red 7 (all +1), the pair counts as zero. You will quickly learn to ignore self-canceled pairs, as well as 8s, 9s and black 7s, since all of these are valued at 0.

When you are good at counting cards in pairs, practice turning them over 3 and 4 at a time. Counting in larger groups really speeds you up, and is a technique professional players use. If you turn over a Ten, 8 with a 2 and black 7, the change in your running count is 0, because the 8 and black 7 aren’t counted at all and the Ten and 2 cancel each other out.

Always strive first to be accurate in your count. Speed without accuracy is worthless. You will do much better in your gambling career if you get in the habit right now of taking the time to learn to do things right.

After you are good at counting cards in pairs and groups of three or four, run through the cards by fanning them from one hand to the other as you count. This technique builds your skill for actual casino play. Allow your eyes to quickly scan the exposed cards for self-canceling pairs, even when these cards are not adjacent to each other.

You should be able to count down a deck in this fashion in 40 seconds or less before you ever attempt counting cards in a casino. Most pros can easily count down a deck in less than 30 seconds. Most professional teams require players to demonstrate that they can count down a deck in 25 seconds or less, with perfect accuracy every time. The legendary card counter Darryl Purpose used to win card-counting contests with his teammates by counting down a deck in 8 seconds flat.

I’ve found that if you can count down a deck in 15-20 seconds or less, you’ll be fine in even the fastest-dealt real-life blackjack games.

No matter how fast you get at counting at home, you will probably find it difficult the first time you actually try to count cards at a casino blackjack table. You may find you forget your running count when you’re playing your hand or talking to the pit boss. In face-down games, you may miss counting some cards as players throw in their hands and dealers scoop them up quickly. You may forget which cards you have already counted and which cards you have not.

Don’t worry about it—every successful card counter has gone through this initial awkward period. You will get better with practice. Before you try counting cards in a casino while actually playing blackjack yourself, spend some time counting while watching others play. Do not sit down to play until you feel comfortable counting while watching the game.

If you expect to play in multiple-deck games, practice counting down multiple decks of cards at home. But be aware that your final running count should go up as you add decks. In a single deck, your final running count should be +2 because of the two red 7s in the decks. But if you are counting down 6 decks, there will be twelve red 7s in the decks, so your final running count should be +12. Multiply the number of decks you are counting by +2 to get the correct final running count for your practice.

Setting Your Starting Count for Casino Play

To use your running count to make betting and playing decisions at the table, you need to know about the “pivot.”

What is a pivot? For the Red 7 count, it’s the running count at which you will know your advantage has risen by about 1% over the game’s starting (dis)advantage. The pivot will be an important indicator in making betting and playing decisions.

If you start with a running count of 0, your pivot will change with the number of decks you are counting, just as your final running count changes with the number of decks.

To keep things simple at the tables, and make your pivot and other indicators the same for all numbers of decks, the easiest thing to do is adjust your starting count.

To make your pivot equal 0 for all numbers of decks, simply multiply the exact number of decks to be dealt by -2 to get your starting running count. (With six decks, you should start your running count at -12. With two decks, you start your running count at -4.)

If you always start your running count in this way, your final count (if you count every card in the deck(s)) should always be 0.

Here’s a simple chart that shows what your running count should start at with various numbers of decks. And yes, there are a few casinos in this world that deal 3, 5, and even 7-deck games. They’re not common, but they exist.

Red Seven Starting Counts

The Red Seven Blackjack Betting Strategy

Once you are proficient at counting, you can begin to apply the Red Seven betting guidelines at the tables. The idea is to raise your bet when you have an advantage over the house, raise it even more when you have more of an advantage, and keep your bet small when the house has the advantage over you.

Remember, when counting cards in a casino, if you always begin your count at the appropriate starting count for the number of decks in play, your pivot is 0. This means, again, that any time your running count is 0, your advantage will have risen about 1 percent over your starting advantage.

This zero “pivot” is a good indicator of when to first raise your bet for nearly all the traditional blackjack games available in this country. About 80 percent of the traditional games have a starting advantage between -0.4 percent and -0.6 percent. So, your zero pivot usually indicates an advantage for you of approximately ½ percent.

This is not a huge advantage. It does not guarantee that you will win the hand—far from it. With a ½% advantage, for every $100 you bet, you will end up in the long run with $100.50, or an extra fifty cents per hundred bucks bet. The important thing is that your count tells you when the edge has shifted from the house to you.

How much should you raise your bet when your running count hits the pivot—or beyond? This depends on many factors, including the rules of the game, the number of decks in play, the penetration (shuffle point), the size of your bankroll, what you can actually get away with in that particular casino, etc. (Casino personnel often view a large betting spread as a sign that a player may be a card counter.)

The chart below will provide a guide for the most common games.

Units to Bet

The general idea is to bet enough when you have the advantage to cover the cost of all the smaller bets you placed when the house had the advantage. Card counters call these small bets “waiting bets.”

Think of the cost of these waiting bets as overhead expenses, or “seat rental,” and you’ll understand why you want to keep these bets small. When the edge shifts to your favor, you want to bet a sufficient amount to cover all these costs, plus make a nice profit.

Card counters call the difference between your waiting bet and your largest bets (placed when you have your strongest advantage) your “betting spread.” For example, if you bet $5 at the top of the shoe, but raise your bet to $10 when the advantage shifts to your favor, and bet up to $40 when your count is highest, indicating your strongest advantage, this would be a 5-to-40 betting spread. You may also express this betting spread as 1-to-8, with a betting “unit” of $5.

The guidelines above are not to be taken as strict betting advice. In many one-deck games, for example, a 1-to-4 spread according to the count will get you booted out in short order, especially if your unit size is $25 or more.

In many shoe games, a 1-to-8 spread would barely get you over the breakeven point. This is why the 0-unit bet is suggested in shoe games at negative counts. It is often impossible to play only at positive counts in shoe games, but it is often wise to leave the table at a negative count.

Many professional gamblers get away with a spread of 1 to 20. They size their top bets according to their bankroll, and get their waiting bets down to the absolute minimum, to maximize their earnings. I myself play with a bet spread even bigger than that.

Note that the suggested bets are in units, not dollars. Your unit size is dependent on the size of your playing bankroll. I’m going to provide some very simple bet-sizing guidelines here that should prove sufficient for most players.

If you intend to take your game further, I recommend my book, Blackbelt in Blackjack : Playing 21 as a Martial Art, which provides very detailed betting advice for those whose careers depend on casino blackjack winnings as a sole or major source of income.

Bet-sizing and bankroll considerations for professional players require a study of standard deviation, normal fluctuations, risk, and the relationship of your advantage to these factors. For now, let’s stick with practical advice that will apply to most recreational players.

The “Trip” Bankroll

It is very important, first of all, for you to define exactly how much money you have available for gambling. Let’s say you go to Las Vegas or Atlantic City a few times per year and you always bring somewhere in the neighborhood of $1,500 to gamble with. Sometimes you win, sometimes you lose, but if you lose it all, no big deal. You’ll be back again in a few months with another fifteen hundred to take another shot at the casinos.

When you go to the casinos, you are always playing with a “trip bankroll.” This is not your life savings, nor are you depending on this money to make your next mortgage payment. This is expendable income to you, earmarked for entertainment.

As a card counter with a “trip bankroll,” you can play very aggressively. Divide your total trip bankroll by 150, and use this as your betting unit. With a $1500 bankroll, you divide:

$1500 / 150 = $10 unit

So, with the betting guidelines above, in the single-deck game you will spread your bets from $10 to $40. In the double-deck games, you’ll spread from $10 to $60. And in the shoe games, you’ll spread from $10 to $80. Whatever the actual size of your trip bankroll, use this method to obtain your betting unit.

If you think these betting guidelines are not aggressive enough for you, please follow my advice and use them anyway, at least until you learn more and get some experience with normal winning and losing streaks.

You will soon discover that even when you play blackjack with an edge over the house, the short-term money fluctuations are huge on your way to the long run, and more aggressive betting than this will often get you into trouble. Even with these guidelines, you will sometimes lose your entire trip bankroll before your trip is over!

In shoe games, with that high bet of $80, you are starting with fewer than 20 high bets with your initial $1500 trip bankroll. That doesn’t give you a lot of wiggle room for bad luck.

The “Total” Bankroll

If the money you intend to go to casinos with represents any significant amount of your total savings, and it is not an easily replenishable amount, then you must size your bets less aggressively. This also means that you must start with a larger bankroll, or play in smaller games, if you want to survive. There are many professional players today who started out with total bankrolls of $5,000 or less, but this is a very tough grind, and often requires a player to (God forbid!) get a job during the toughest times.

As a general rule, if your card-counting bankroll is not replenishable, obtain your unit size by dividing your total bankroll by 400. Then use the same betting chart above to size your bets. Serious players will need to use much more precise betting strategies, according to their advantage, table conditions, the necessity for camouflage, etc. Again, those with professional aspirations should see Blackbelt in Blackjack for an in-depth treatment of this subject.

The Red Seven Blackjack Playing Strategy

Using the Red Seven Count, you can also increase your advantage over the house by deviating from basic strategy according to your running count. First of all, insurance is the most important strategy decision. In single-deck games, assuming you are using a moderate betting spread, insurance is almost as important for a card counter as all other strategy decisions combined.

Conveniently, you have a very nice insurance indicator with the Red Seven Count. In 1- and 2-deck games, you simply take insurance any time your running count is 0 or higher. In all shoe games, take insurance at +2 or higher.

As for other playing decisions, there are only a few to remember. Any time you are at 0 or higher (any number of decks), stand on 16 vs. 10 and on 12 vs. 3. (According to basic strategy, you would hit both of these.)

In single-deck games, the 16 vs. 10 decision is the second most important strategy decision for a card counter—insurance being first. After you find these strategy changes easy, there are a couple of others you can add that will increase your advantage a bit more. At running counts of +2 or higher, with any number of decks, stand on 12 vs. 2 and on 15 vs. 10; and double down on 10 vs. X.

In multi-deck games, by using this simple running count strategy, you will be taking advantage of about 80% of all possible gains from card counting. Using the simple Red Seven Count, you have no strategy tables to memorize. You simply have basic strategy, which you play on more than 90% of your hands, and a few changes that you will make according to your running count.

In my opinion, most card counters would be wise to ignore more difficult strategies because of the cost of mistakes if you are not perfect in deploying them. Any system that slows you down, or causes mental fatigue or errors, will put more money into the casinos’ coffers than your pockets. Don’t be tempted by a system just because it works better on paper. The simple Red Seven Count works at the casino tables, and it gets the money. That’s the goal.

However, if you find yourself interested in using a more advanced card counting system to take advantage of every possible gain available from counting, I recommend that you look at the Hi-Lo Lite Count or the Zen Count, both included in Blackbelt in Blackjack . There is also an advanced version of the Red Seven Count in that book that is stronger and more versatile than the simple version presented here.

Blackjack Table Conditions

The actual overall advantage that a card counter can get over the house depends primarily on how deeply into the deck the dealer is dealing between shuffles. Card counters call the depth of the deal “penetration.” The deeper the penetration, the more often you’ll see counts that indicate you have an advantage and the stronger the advantage will be.

Without deep enough penetration, you will find that you simply count down shoe after shoe without seeing any high counts. The worse the penetration, the bigger your betting spread has to be to overcome all those extra waiting bets. If the penetration is 50% or less, you’re wasting your time counting cards in that game.

Card counting is also unlikely to be profitable in any game where blackjacks pay less than the traditional 3:2 payout. That’s because you must overcome a higher starting disadvantage on these games. In a game with a 6:5 payout on blackjacks, for example, you must overcome an additional 1.4% house edge. To overcome this, you must use an enormous betting spread.

For more information on table conditions and your overall edge from counting cards, see Blackbelt in Blackjack.

History of the Red Seven Count

I first published the Red Seven Count in 1983. It was very controversial when published, as many experts believed it impossible to whittle a system down to the bare basics, require no math whatsoever at the tables aside from the counting itself, and still get any significant edge over the house. Since that time, however, many independent computer simulation studies have shown the Red Seven Count to be exactly as I first described it, a professional-level system that is both easy and powerful.

Numerous system developers since have used the same approach I pioneered, but none have matched both the simplicity and power of the Red Seven. Some authors, like Ken Uston and George C., developed slightly more powerful systems using my unbalanced point count theory, but their systems are also more difficult to use than the Red Seven. Others, like rocket scientist Olaf Vancura and Ken Fuchs, developed the Knockout Count, very similar to the Red Seven Count, that perform similarly – sometimes weaker, sometimes stronger – in most game conditions. ♠


For more information on card counting on other methods professional gamblers use to win at blackjack, see Arnold Snyder’s Blackbelt in Blackjack. For complete information on the game of blackjack, including its history, variations, and stories of its great players, see The Big Book of Blackjack by Arnold Snyder.