Knock-Out Blackjack: Excerpts

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A book by Olaf Vancura and Ken Fuchs

Excerpts from Chapters 1 and 5. ©1996, Olaf Vancura.
Note:  This book was updated in 1998.

Thorp’s book eventually created a sensation, riding a wave of popularity to the New York Times Bestseller List. And why not? For, as Bryce Carlson says in Blackjack for Blood1 “Here was a book, written by a respected mathematician from a prominent university, that contained the secret formula for making free money — or so some thought.”

Blackjack’s popularity soared. Major magazines ran articles describing how the game could be beaten. Consider the accolades in the following excerpt from Life: “Thorp does not cheat. But Thorp cannot lose. Humans have been betting on games of chance since the dawn of history, but Thorp must be considered the greatest system player of all time.”

Not surprisingly, casinos first reacted with anxiety. Comments Carlson, “If the man in the street was overreacting a bit–it was nothing compared to the hysteria that seized the casino industry. Suddenly, they had nightmares of thousands of trained counters swooping down on them like swarms of merciless locusts, devouring every hundred dollar bill in sight.” This frame of mind led many casinos to change the rules of the game to make it harder for players to win.

This situation did not last long, as Lance Humble and Carl Cooper comment, in The World’s Greatest Blackjack Book2, “Too many players refused to play blackjack with these unfavorable rules, and the win volumes dropped off dramatically.”

To bring back players, casinos were soon forced to capitulate and reinstate the old rules. After some tense days, the casinos began to relax when they realized that though many people thought they could beat the game, few actually could. Remark Humble and Cooper, “The casinos soon realized that they had nothing to fear. The publicity that Thorp’s book provided turned out to be a boon.” Why? “The players kept losing at exactly the same rate as before, only now there were more of them.”

Blackjack’s popularity continues in high stride still today. Apparently the lure of playing a game that is capable of being beaten is sufficient for most people, who somehow believe that Dame fortune will smile on them despite their inadequacies in play.

But these people seem to miss the point. It is not enough merely to show up to a game that can be beaten. One needs to play accordingly. It’s similar to a college midterm exam where you want to get an A grade. To show up and take the exam is necessary, but obviously not sufficient for the A. You must not only show up; you must do the preparatory work beforehand.

Rather than put in the work, practice, and patience (!) to learn to play well, Carlson argues, “This is not what the people were looking for. They preferred instead to continue in their old, uninformed ways.” Humble and Cooper opine that most people who bought Thorp’s book simply did not take the time to master card-counting.

Of course, it didn’t help when supposed “experts” burst on the scene with misinformation. The game of blackjack had, in one form or another, been around for about a century. And during this time it had been played very poorly by players. Suddenly, when the diligent work of an esteemed scientist became public, out of the gaming industry woodwork came self-proclaimed card-counting experts, ready to profess that everyone else knew nothing!

It’s amazing how that seems to work, isn’t it? This is reminiscent of the developments in the first part of this century shortly after Albert Einstein published his general theory of relativity. The great scientist Arthur Eddington was in West Africa observing an eclipse in an attempt to perform a test of Einstein’s ideas. Queried a reporter to Eddington, “It is said that the general theory is so complicated that there are but three men in the world who can understand it.” Eddington replied, “Who’s the third?”

Unfortunately, some in the casino industry apparently didn’t understand the concepts and thus misrepresented the ideas. John Scarne boldly wrote, “You can’t remember all the exposed cards dealt to a full table of players.” While the statement may be true for most people, it is quite irrelevant and thus misleading. The Thorp system didn’t require anyone to remember all the exposed cards. Quite the contrary, Thorp recognized that this was beyond the means of mere mortals and designed a system accessible to diligent everyday folks.

Furthermore, Wilson comments that writers associated with casino management may “purposely give some bum steers.” Scarne wrote of card-counting as “chicanery” and that “an expert job of card casing requires unusual natural ability.” Sentiments such as these, from purported gaming “experts” with ties to the gaming industry, seemed intended only to unfairly dampen the impact of Thorp’s findings.

But even Thorp, who knew that his book would release the genie from the bottle, did not realize the future impact of his work. The fact that a respected scientist had shown that blackjack could, in principle, be beaten was enough to move casinos to action. Thorp wrote, “Eventually, when the strategy we outline becomes general practice, casinos may change the game or discontinue it.”

Clearly it is true that the game has changed markedly. In the 1960s, casinos typically dealt from a single deck dealt through to the end. Whenever the pack was exhausted, often mid-hand, the dealer would reshuffle the discards and continue dealing. The burn card was shown to all players. Today, typically 1/4 of the pack is cut off and these cards are never brought into play until perhaps the next shuffle. In fact, it is a cardinal sin for a dealer to now run out of cards; in some casinos this is cause for immediate dismissal.

But Thorp greatly underestimated his influence over the industry. He surmised, “This (change) will happen gradually, over a period of years, and even then, it will not take place in all of the many and diverse places where blackjack is played.”

Three Sides of the Same Coin
The old adage that there are two sides to every story certainly applies to recaps of the ensuing events after the publication of Beat the Dealer. Below is a sampling of spin (or perhaps revisionist history) provided by the feuding sides in the casino/card counter battle (underlined emphasis courtesy of your authors).

The card counter’s side: Carlson writes, in Blackjack for Blood, “Their paranoia out of control, the Las Vegas casinos snapped! On April Fool’s Day 1964, the casinos on the Las Vegas Strip changed the rules of blackjack. The first (and only) time the rules of a major casino game had ever been significantly altered. And the changes were drastic. Doubling down was restricted to two-card totals of 11 only, and a pair of Aces could no longer be split. The effect on the average player was disastrous, and play at the tables all but vanished.

The casino’s side: A “suave PR man,” in a Newsweek article dated April 13, 1964, indicates that the operators eliminated the “fringe benefits” of the game, namely “the right to double most bets and to split hands of two Aces.”

A third side to the story? Thorp’s nonchalant reply (in the same Newsweek article) to all the hoopla: “Instead of five hours, now I’ll have to play seven to make the same money.”

The most popular counts today are, perhaps surprisingly, variations of the original point-count systems. For all the improvements and simplifications that have occurred in the last 35 years, card-counting still remains relatively inaccessible to most of us. It’s simply too much to do, what with all the bells clanging, dealers talking, cocktail waitresses jiggling, pit bosses staring, eyes-in-the-sky watching, players chatting, and money changing hands.

So have we exhausted the possibilities? Has blackjack been milked for all it’s worth? Thankfully no. Is there anything left for the rest of us? The answer is yes!

Necessity is the mother of invention, and the need for simplicity is why the Knock-Out method was conceived. We believe the Knock-Out system is, to date, the easiest professional-level card-counting system. It is the system capable of making better players out of almost all of us. The ease of play makes card-counting a “fun” event for even the casual player. And the simplicity does not come at the expense of performance. The K-O system is a top caliber tool at your disposal. Let’s get ready for round 2.

Ease of Use

The unbalanced nature of the Knock-Out count (which eliminates the need for remaining deck estimation and a true count conversion), the natural level-1 card values, the lack of any side counts, the innovation of a reduced and rounded matrix (which eases the effort to memorize and apply the strategy matrix), and the ability for you, as a player, to custom tailor the Preferred system, makes the K-O system unique. To date, no other system has all, or even three, of these attributes.

The following table compares the most popular systems today in terms of ease of use. The three comparison parameters left of the solid bar (type, level, and whether or not a side count is employed) are intrinsic properties for each particular system and are therefore fixed3.

The comparison parameter to the right of the bar (whether or not a rounded matrix exists) is included for the present-day version of these systems. In principle, any system developer so inclined could develop a rounded matrix for a system, hence this is not an intrinsic property of the system per se. But of course this requires effort, whereas the K-O system has already been designed with ease of use in mind.

System Comparison: Ease of Use
Red 7U1+*NN
Hi-Opt IB1YN
Omega IIB2YN
*Red 7 requires the player to keep track of colors on 7-valued cards.


We’ve demonstrated that the K-O system is easy to apply. But how well does it perform? Below we give comparison tables to demonstrate the power of the K-O system.

We note that our rounding of the K-O matrix comes at the expense of expectation. In the “Reduced” comparisons which follow, we will always be adopting the top 16 plays of the K-O Preferred system. Thus, we are comparing the Knock-Out rounded matrices to the unrounded top 16 matrix plays of the other systems, in effect giving the other systems the benefit of a slight edge.

Let’s first consider a 2-deck game with the benchmark rules. We choose the 2-deck game to study in detail because it is a compromise between single-deck and multiple-deck games. To remind you, our benchmark 2-decker included the dealer standing on soft 17 (S17), doubling down on any first two cards (DOA), no doubling after splitting (noDAS), and no surrender. We have fixed the penetration at 75%4.

Below is a table which summarizes the 2-deck results. Each entry in the following table is based on a simulation of at least several hundred million hands. As before, perfect play was assumed; no betting or playing errors were introduced. To be fair to each system, we have placed the different systems on the same scale in a modified proportional betting fashion, as described in the last chapter. Note that by doing so, we force each system to have nearly the same chance of ruin5.

We have adopted the benchmark of the last chapter; each entry was calculated based on a spread of 1 to 5 units. As before, intermediate values rise linearly with the player’s expectation, with a constant of proportionality (ramp) of 3, such that the minimum wager is 1 unit, and the maximum wager of 5 units is made at an expectation of +1.67%6.

Simulation Results: 2-deck game (7)
(DOA, noDAS, S17, 75% pen;
Spread 1 to 5 w/ ramp of 3)
Red 70.821.081.12
UZ II0.851.16N/A
Hi-Opt I0.801.091.19
Omega II0.851.151.28

In constructing this table and those that follow, we have sought to place the systems on an even playing field. As such, each system’s variation (of Core, Reduced, or Full) has approximately the same number of strategic plays to memorize that potentially deviate from the basic strategy:

The “Core” column, for each system, assumes that the only strategy deviation from basic strategy is the insurance wager.

The “Reduced” column includes only the 16 most significant matrix entries when comparing to the basic strategy. For the unbalanced systems (which behave very similarly strategically), we have adopted the 16 positions which exist in the Preferred form of the Knock-Out system (of course each system has different numerical values as appropriate for the 16 matrix positions). For the Knock-Out 6- and 8-deck simulations, only 14 plays are used (the C value is omitted). For the balanced systems, we have adopted Don Schlesinger’s “Sweet 16” set of 16 plays which are the most rewarding to memorize.

The “Full” column includes the 40 or so most significant matrix entries for all systems (much less than 40 entries for the K-O and unbalanced systems in the case of 6 and 8 decks; see Appendix III).

As you can see, the systems’ expectations are all bunched together fairly tightly. No system’s performance stands out as superior over all the rest.

Note in particular that K-O compares admirably with all other systems. Indeed, the two most popular systems in use today, the Hi-Opt I and the High-Low, are both edged slightly by the Knock-Out system. This is despite the fact that the Knock-Out system is vastly simpler to employ.


1 Bryce Carlson, Blackjack for Blood, 1994, CompuStar Press.
2 Lance Humble & Carl Cooper, The World’s Greatest Blackjack Book, 1980, Doubleday.
3 The side count, in principle, can be discontinued by a player. Since side counts require so much effort to implement, some may argue that a proper comparison between systems thus should be made without any such extra counts. Qualitatively, the elimination of side counts (for those systems which employ them) comes at considerable expense in expectation. In our comparison, eliminating all side counts would serve to enhance the relative performance of K-O and other singular-count systems.
4 See Appendix IV for a discussion of the effect on K-O of varying penetration.
5 See Appendix V for confirmation of this statement.
6 As mentioned in the last chapter, varying the ramping factor has little effect on the relative performance of the systems.
7 Sources for all comparisons in this chapter– Red 7: Core adapted from Arnold Snyder’s Blackbelt in Blackjack; Reduced and Full plays for 1 and 2 decks adapted from Arnold Snyder’s “The Big Tilt” article in Blackjack Forum, March, 1994; Reduced and Full plays for 6 and 8 decks based on applicable extrapolations from K-O. Unbalanced Zen II: All versions adapted from George C’s The Unbalanced Zen II. Hi-Opt I: Core and Full adapted from Lance Humble and Carl Cooper’s The World’s Greatest Blackjack Book; Reduced plays are Humble and Cooper’s matrix entries for Don Schlesinger’s “Illustrious 18” less the two splitting Tens plays, hereinafter referred to as the “Sweet 16.” High-Low: Core and Full adapted from Stanford Wong’s Professional Blackjack; Reduced plays are Wong’s matrix entries for Schlesinger’s “Sweet 16.” Omega II: Core and Full adapted from Bryce Carlson’s Blackjack for Blood; Reduced plays are Carlson’s matrix entries for Schlesinger’s “Sweet 16.”

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