A System for Beating the Over/Under Side Bet

by Arnold Snyder

© 1989 Arnold Snyder

Introduction

Much of this report will seem too technical for the average player. If you don’t quite grasp some of the mathematical concepts, don’t worry about it. The over/under card counting system described is simple and easy to apply. Just follow the recommendations and you should have no trouble. The more technical explanations are for the math-heads who will hopefully appreciate the full story. –A.S.

The Over/Under Rule

At the time of this writing (August, 1989) the only U.S. casino that offers the over/under rule at its blackjack tables is Caesars Tahoe in Stateline, Nevada. [Editor’s note: As of 2006, the Over/Under rule is not offered in the U.S. However, it is offered in casinos in Eastern Europe, Asia, and elsewhere.] Caesars has a two-year exclusive contract with the developer of this rule variation to offer this game in the U.S. In the summer of 1990, the developer may market this rule variation to other U.S. casinos.

Note: Technically, the “rule” of a game cannot be protected by copyright or patent. However, the over/under bet requires an alteration of the blackjack table layout in order to facilitate the side bet. The developer, Mr. Ken Perrie, a Caesars’ pit boss, owns the rights to the design of his unique table layout.

Bets on the regular blackjack hand are placed in the large central circle (the normal betting spot). Optional over/under side bets may be placed in the small circles to the left and right of the center circle. The side bet circles will be labelled “over” and “under.”

How the Over/Under Bet Works

As with the normal blackjack bet, over/under bets must be placed prior to the deal. Over/under bets are always optional side-bets. No player at an over/under game is ever required to place an over/under bet.

The player who places a bet in the over circle is betting that his first two cards will total more than 13. The player who places a bet in the under circle is betting that his first two cards will total less than 13. These are even money bets. Aces always count as one when determining totals for the over/under bet. King, Queen, Jack and 10 all count as ten. All other denominations count as their face values.

Over/Under Restrictions

No player may place an over/under bet unless he is betting on a regular blackjack hand. The amount bet on over or under may never exceed the amount bet on the blackjack hand. At Caesars Tahoe at this time, no over or under bet may ever exceed $100.

All over/under bets are settled prior to the play of the blackjack hands. The outcome of over/under bets have no effect on any aspect of the blackjack game.

Analysis of the Over/Under Game

Compared to regular blackjack, the over/under game is easy to analyze. By considering all possible two-card totals, it is a straightforward calculation to count how many of these hands total over 13, and how many total under 13.

If we assume that the player will bet one unit on every hand that his hand will total over 13, then compare the results of his wins and losses, we see that he will lose about 6 1/2% of his total amount bet. If he always bets on the under side, he will lose slightly more than 10% of his total under bets. The house obtains its advantage from the fact that the house always wins over and under bets on player totals of exactly 13.

So, “basic strategy” for the over/under bets is to never bet on either side. The house has the advantage either way in the long run.

However, it is also obvious that this house advantage will change throughout the course of the game. An excess of high cards remaining to be dealt would make a hand totaling over 13 more probable. And an excess of low cards will make an under 13 hand more probable.

In order to devise an optimal card counting system that will tell us when the over or under bets have become advantageous, we must first figure out the effects of removing each of the individual cards. Again, this is a very straightforward mathematical calculation. It is done exactly like the initial calculation we did to determine the house advantage off the top of the six-deck shoe [editor’s note: analysis of 1, 2, 4 and 8 deck Over/Under games is included in the supplement at the end of this report], except that we remove one of each denomination of card in turn, refiguring the outcome on the over and under bets.

We will find, for instance, that when we remove one ten-valued card, we come up with slightly fewer possible two-card hands that will total over 13, and slightly more two-card hands that will total under 13. The difference between the house advantage off the top and minus one ten tells us the effect of removing a ten. Note that each card will have two effects of removal—one effect on the over bet, and a different effect on the under bet.

These are the approximate effects of removal in percent for each denomination of card in the 6-deck over/under game (for effects of removal in 1, 2, 4, and 8-deck games, see the supplement at the end of this report):

Effects of Removal in the 6-Deck Over/Under Game

	A	2	3	4	5	6	7	8	9	10
Over:	+0.60	+0.60	+0.60	+0.20	+0.11	+0.01	-0.09	-0.19	-0.29	-0.39
Under:	-0.71	-0.71	-0.31	-0.21	-0.11	-0.01	+0.08	+0.18	+0.28	+0.38

These effects are much greater than those for regular blackjack. The removal of one ace, for instance, in a 6-deck blackjack game, has an effect of approximately -0.10% on your regular blackjack hand. Removing an ace has 6 times this effect on any over bet, and seven times this effect on an under bet.

It is easy to see why the ace and deuce, for instance, have such high effects of removal. Consider that if either one of your cards is an ace or a deuce, you could not possibly lose the under bet, nor could you possibly win the over bet.

Also note that these effects of removal are for removing just one card from the top of a 6-deck shoe. In regular blackjack, effects of removal are usually given for removing one card from the top of a single deck. The effect of removing an ace from a single deck is approximately -0.6% on your regular blackjack hand. If the over/under bet were allowed in a single-deck game [editor’s note: 1, 2, 4 and 8-deck games with the Over/Under rule later became available], the effects of removing one ace would be -4.40% on the under bet, and +3.73% on the over bet.

In other words, even though the house advantage over you on the over bet in a single-deck game is about 6.8%, the removal of just one ace and one deuce in a single-deck over/under game would put the edge in your favor by almost 0.7%! Although there are no single-deck over/under games at this time, the volatile swings in advantage caused by the effects of removal make even the 6-deck game a profitable game for card counters. Removal of a single ace, deuce, or three, for instance, from the 6-deck shoe has more effect on your over/under betting opportunities than the removal of any of these cards would in a single-deck regular blackjack game.

Devising an Over/Under Card Counting System

First let’s look at how well some common blackjack card counting systems correlate to the over/under bet:

Correlation of Various Card Counting Systems to the Over/Under Bet

COUNT SYSTEM	A	2	3	4	5	6	7	8	9	10	UND	OVR
EINSTEIN/HI OPT I
	0	0	1	1	1	1	0	0	0	-1	-0.56	0.64
DHM/GORDON
	0	1	1	1	1	0	0	0	0	-1	-0.74	0.79
GRIFFIN-1
	0	0	0	1	1	1	1	0	0	-1	-0.46	0.46
BRAUN +-/DUBNER/HI-LO
	-1	1	1	1	1	1	0	0	0	-1	-0.50	0.57
USTON +-
	-1	0	1	1	1	1	1	0	0	-1	-0.32	0.41
REVERE ADVANCED +-
	0	1	1	1	1	1	0	0	-1	-1	-0.73	0.78
CANFIELD EXPERT
	0	0	1	1	1	1	1	0	-1	-1	-0.55	0.62
ITA GREEN FOUNTAIN
	-1	1	1	1	1	1	1	0	-1	-1	-0.50	0.56
HI OPT II/STEPPINE
	0	1	1	2	2	1	1	0	0	-2	-0.64	0.67
ANDERSON/REPPERT
	-2	1	1	1	0	1	1	0	-1	-1	-0.30	0.38
R&T POINT COUNT
	0	1	1	2	2	2	0	0	0	-2	-0.63	0.66
REVERE POINT COUNT
	-2	1	2	2	2	2	1	0	0	-2	-0.42	0.50
CANFIELD MASTER/GRIFFIN-2
	0	1	1	2	2	2	1	0	-1	-2	-0.64	0.66
WONG’S HALVES
	-1	0.5	1	1	1.5	1	0.5	0	-0.5	-1	-0.44	0.51
GRIFFIN-3/USTON ADVANCED
	0	1	2	2	3	2	2	1	-1	-3	-0.61	0.65
REVERE ADVANCED (’71)
	-4	2	3	3	4	3	2	0	-1	-3	-0.38	0.46
REVERE ADVANCED (’74)
	0	2	2	3	4	2	1	0	-2	-3	-0.68	0.71
ZEN
	-1	1	1	2	2	2	1	0	0	-2	-0.51	0.55

Note that the most common counting systems correlate poorly. Hi-Opt I has about a 64% correlation on the over bet, and about 56% on the under bet. The High-Low is worse; 57% over and 50% under. The Zen Count’s about the same. The Uston +/- is terrible. The best normal card counting system for this game is the DHM (a mail order card counting system of years gone by, no longer in print.) Second best is the Revere Advanced +/-, another ancient system no longer in wide usage. Third best is the 1974 Revere Advanced Point Count, which correlates 71% on the over bet and 68% on the under bet. But this is another system that is not widely available, and a difficult level 4 strategy that would be unwieldy for most players.

Considering the fact that the effects of removal indicate to us that over/under betting opportunities provide far more profit potential in this 6-deck game than regular blackjack, let’s devise some card counting systems specifically aimed at attacking the over/under. Here are six possible over/under counts:

Six Possible Over/Under Counts

COUNT	A	2	3	4	5	6	7	8	9	10	UND	OVR
Count 1	1	1	1	1	0	0	0	0	0	-1	-0.90	0.92
Count 2	1	1	1	1	1	0	0	0	-1	-1	-0.89	0.91
Count 3	1	1	1	1	1	1	0	-1	-1	-1	-0.86	0.88
Count 4	2	2	2	0	0	0	0	-1	-1	-1	-0.94	0.97
Count 5	2	2	1	1	0	0	0	-1	-1	-1	-0.98	0.96
Count 6	2	2	2	1	1	0	0	0	0	-2	-0.91	0.94

We can see here that Count #4 and Count #5, both level two systems, correlate the best. My choice of system, however, is Count #1, for a number of reasons. First of all, as a level 1 system, it’s the easiest to learn and use. Second, if I assume that your are going to be using your over/under count not only to make over/under betting decisions, but also to play your regular blackjack hand, Count #1 is the best all-purpose card counting system. It has an insurance efficiency of 85%, the same as Hi-Opt I, and a playing efficiency of 47%, not that much worse than the High-Low’s 51%. For convenience, let’s call Count #1 the “Over/Under Count.”

The Over/Under Count

	A	2	3	4	5	6	7	8	9	10	UND	OVR
The Over/Under Count	1	1	1	1	0	0	0	0	0	-1	-0.90	0.92

Although it is possible to develop many multi-parameter approaches that would optimize potential profits on both the over/under bets and the regular blackjack hand, I would not consider these more difficult strategies to be worth the trouble. Having a system with a high betting correlation for your regular blackjack hand is useless. The over/under bets are much more important. And to strive for a playing efficiency of 60%, or even 70%, in a 6-deck game, is a waste of time. The potential strategy gains are extremely small.

The vast majority of your profit potential will come from betting accurately on your over/under opportunities. You only want to be and play the regular blackjack hand well enough to reduce the ½% house advantage to as near a break even point as possible. You can’t table hop over/under games. You’re likely to miss betting opportunities on the under side when the count is negative.

For extreme ease, if you are already using a balanced level 1 card counting system, such as Hi-Opt I or the Hi-Lo count, you could use the same strategy tables to play your hand using the Over/Under Count, and you will be playing your regular blackjack hand with great accuracy. This is a count-per-deck strategy chart drawn up specifically for using the Over/Under Count to play your blackjack hands.

Count Per Deck Strategy Chart for the Over/Under Count

	2	3	4	5	6	7	8	9	10	A
16							+5	+3	0	+3
15									+5
14	-4
13	0	-2	-3	-4
12	+4	+2	+1	-1	-2
A7									+5	0
11								-4	-4	-1
10							-5	-2		+3
9	+1	-1	-4			+4
8				+4	+2
Insurance: +3

The next step in analyzing this game is to figure out the value of each point-per-deck; in other words, how much does your (dis)advantage change with each true count? The value of a point is a function of the effects o9f removal and the assigned point values. Since we have different effects of removal for the over bet, the under bet, and the regular blackjack hand, each true count will technically have three different values.

If you have a background in statistics, you may calculate the value of a true point for each of the three possible bets by dividing the inner products of the effects of removal and their respective assigned point values by the sum of the squares of the point values. Note that you should first multiply the 6-deck effects of removal for the over and under bets by 6, in order to obtain the effects of removal per deck. The effects of removal for the regular blackjack hand are in Peter Griffin’s Theory of Blackjack, as well as numerous other publications. Griffin also explains how to find the inner product.

Fortunately, Sam Case has a computer program that spits out this data in a fraction of a second. You don’t have to do the math, or even understand it. Just trust me that each true count using the Over/Under Count described above has the following values:

Over/Under True Count Values

Over bet:	2.70%
Under bet:	2.58%
Regular blackjack hand:	0.35%

Do note that each true count with this system affects your regular blackjack advantage by only -.35%, instead of the 0.5% we usually expect with level one systems. This is due to the lower betting correlation, caused primarily by assigning a plus value to the ace. We are willing to relinquish the betting accuracy on our blackjack hand since the over/under bets offer so much more profit potential.

Caesars Tahoe’s 6-deck games use downtown Vegas rules with double after splits. The player’s starting advantage in this game is -0.54%. The following chart shows your expectation in percent on each of the three possible bets you may make at various true counts.

6-Deck Expectation for Various True Counts

True Count	Blackjack Bet	Over Bet	Under Bet
-9	-3.69	-30.85	+13.15
-8	-3.34	-28.15	+10.57
-7	-2.99	-25.45	+7.99
-6	-2.64	-22.75	+5.41
-5	-2.29	-20.05	+2.83
-4	-1.94	-17.35	+0.25
-3	-1.59	-14.65	-2.33
-2	-1.24	-11.95	-4.91
-1	-0.89	-9.25	-7.49
0	-0.54	-6.55	-10.07
+1	-0.19	-3.85	-12.65
+2	+0.16	-1.15	-15.23
+3	+0.51	+1.55	-17.81
+4	+0.86	+4.25	-20.39
+5	+1.21	+6.95	-22.97
+6	+1.56	+9.65	-25.55
+7	+1.91	+12.35	-28.13
+8	+2.26	+15.05	-30.71
+9	+2.61	+17.75	-33.39

Now, we’re ready to devise a strategy for attacking this game. One of the nice features about the over/under bets is that you never have to place them. This means that all of the negative advantages that occur on these bets at various counts will have no effect on your expectation since you will not be betting. So, let’s simplify the above chart to show the effective player/house advantages at each true count, assuming that the player only bets on over or under when the player has the edge.

It can also be shown that if we place two equal simultaneous bets on events with different expectations, our combined expectation is the same as if we had placed one bet of the total amount on the average of the two expectations. For instance, the expectation on your blackjack hand at a true count of +3 is +0.51%. The expectation on the over bet is +1.55%. Therefore, if I place a $100 bet on each of these, I would expect to profit in the long run 51 cents on my blackjack hand and $1.55 on the over bet. This is a total profit of $2.06 on the $200 of total action. $2.06 is 1.03% of $200. 1.03% is also the average of .51% and 1.55%.

This means that we can further simplify the chart of our expectations at various true counts by averaging our advantages when we place simultaneous bets. The following simplified chart shows your actual expectations at various true counts when you place over or under bets when you have the advantage.

Expectation at Various Over/Under Count True Counts with the Over/Under Bet

Count	% Expectation
-9	+4.73*
-8	+3.62*
-7	+2.50*
-6	+1.39*
-5	+0.27*
-4	-0.85*
-3	-1.59
-2	-1.24
-1	-0.89
0	-0.54
+1	-0.19
+2	+0.16
+3	+1.03*
+4	+2.56*
+5	+4.08*
+6	+5.61*
+7	+7.13*
+8	+8.65*
+9	+10.18*
* Combined blackjack and over/under bets

Now you can see why this is such a nice game for card counters. Not only does your advantage go up on both positive and negative counts—and quite a bit higher than a card counter’s advantage every rises—but the house never has an advantage over you of much more than 1 ½%.

We can now analyze your profit potential in this game using standard 6-deck frequency distributions. The frequency distributions for this count are identical to those for the Hi-Opt I counting system.

I visited Caesars Tahoe in June of 1989, and again in August, in order to play the over/under games and observe the shuffling standards. There is a wide variance in deck penetration on Caesars’ 6-deck games. On an average, most dealers deal out about 4 ½ decks, or 75% between shuffles. During both of my visits, however, I was able to find dealers who dealt out 5 decks. You will also see dealers who do not deal out much more than 4 decks.

As with any blackjack game, deep penetration is important for card counters. These are frequency distributions, in hands-per-hundred for the Over/Under Count, assuming 4 decks dealt out, 4 ½ decks dealt out, and 5 decks dealt out.

Over/Under Count Frequency Distributions

True Count	4 Decks Dealt	4.5 Decks Dealt	5 Decks Dealt
-9	0	0	0.5
-8	0	0.5	0.5
-7	0.5	0.5	1
-6	0.5	1	1.5
-5	1	1	1.5
-4	2	2	2.5
-3	3	3.5	4
-2	8	7.5	7
-1	14	14	13
0	42	40	37
+1	14	14	13
+2	8	7.5	7
+3	3	3.5	4
+4	2	2	2.5
+5	1	1	1.5
+6	0.5	1	1.5
+7	0.5	0.5	1
+8	0	0.5	0.5
+9	0	0	0.5

If you are mathematically inclined, you may use these frequency distributions to estimate your potential advantage and win rate in dollars with various betting approaches. My analysis yields the following data.

Flat Betting

For the over/under game, we’ll define flat betting as placing the same size bet on all of your blackjack hands, but also placing a bet of the same amount on either the over or under only when it is advantageous to do so. Using the Over/Under Count, this means placing an over bet when your true count is +3 or more, and placing an under bet when your true count is -4 or less.

Flat betting is not a good approach to this game, though it is moderately favorable when the penetration is deep. If only 4 decks are being dealt out, your expectation is -0.1%. If 4 ½ decks are dealt out, your expectation is 0.1%.

In other words, with normal penetration, you will be playing a break even game. If 5 decks are dealt out, your expectation goes up to +0.4%. With $100 bets, and 100 hands per hour, this would be a potential win rate of about $50 per hour. If you have a large bankroll and you can afford the potential fluctuation, this approach may appeal to you—especially if you’re playing for comps. Generally, I do not advise serious players to risk so much for so little. It is remarkable that you are able to get any advantage at all flat-betting a 6-deck game, playing through all negative counts.

1-to-2 Spread

You bet one unit on your blackjack hand when the true count is between -5 and +2. Below -5, or above +2, you bet 2 units. At -4 or below, you match your blackjack bet with an under bet. This means you will be betting one unit on under at -4 and -5, and two units on the under at -6 or below. At +3 or above, you match your two unit blackjack bet with a 2 unit over bet.

With 4 decks dealt out, your expectation is +0.2%, nothing to write home about. With 4 ½ decks dealt out, you’ll expect to win at the rate of about 0.5%. With 5 decks dealt out, you’ll have a full 1% expectation. Spreading from $50 to $100, this would make you about $75 per hundred hands.

1-to-4 Spread

I analyzed this spread with a 1 unit bet on the blackjack hand from -4 to +1; 2 unit blackjack bets at -5 and +2; 4 unit blackjack bets below -5 and above +2. Under bets match the blackjack bets (1, 2 or 4 units) at -4 and below. Over bets match the blackjack bets (4 units only) at +3 and above.

With 4 decks dealt out, this spread will get you an advantage of about 0.7%, not bad for the poor shuffle point. 4 ½ decks dealt out will get you a 1.1% advantage. And with 5 decks dealt out, your expectation is about 1.6%, or about $88 per hundred hands.

You may have noticed that the percentage advantage seems to be climbing with larger spreads faster than the $ expectation. For instance, flat-betting $100 chips with 5 decks out provides a 0.4% advantage, and about $50/hour. Using a 1-to-4 spread, your advantage climbs to 1.6%, but $ expectation is only $88/hour.

The reason for this is that we are obtaining the larger spread by lowering the minimum bet, as opposed to raising the maximum. Since the maximum bet allowed is $100, I’m analyzing the maximum potential dollar return for players who can afford it. Smaller stakes players would find that their $ win rate advanced as expected if they raised their top bets. For instance, flat betting $25 chips, with 5 decks dealt out, would net about $12 per hour. Spreading from $25 to $50 would get you about $38 per hour. And spreading from $25 to $100 would take in about $88 per hour.

I’m analyzing this game with conservative spreads so that you can see that it is not difficult to get the best of this 6-deck game, even with a small spread. Remember, if you’re barred at Caesars Tahoe, there are no other casinos in this country that offer over/under games at this time. Brave fools who either have, or believe they have, incredible acts, may use the frequency distributions provided to discover that they will have no trouble killing the game, even with only 4 decks dealt, with a spread of $5 to $100. Good luck!

Players with substantial bankrolls, who are put off by the limitations of the $100 max bet, may be tempted to analyze the potential win rates if you spread to multiple hands when the over/under bets are favorable. Such tactics will greatly increase your expectation. If you are spreading from $25 to $100 on one hand, using the 1-to-4 strategy described above, and playing a second hand only when the true count is at +3 and above, or -6 and below, your expectation would be about 2.2%, and about $190/hour.

Over/Under Count Bankroll Requirements

One of the nice things about the over/under bet is that it is a true side bet. Betting $100 each on your blackjack hand and the over bet, simultaneously, is not the same thing, as far as fluctuations go, as betting $200 on your blackjack hand, or even playing two simultaneous blackjack hands. The reason for this is that the over/under bets are not affected by the dealer’s upcards.

When you are playing simultaneous blackjack hands, you must account for the covariance. In other words, if the dealer gets a total of 20, both of your hands are going up against this strong total. If the dealer gets a natural, neither of your hands win money. But over/under bets are not tied to the dealer’s hand in any way. If you place an over bet, and you are dealt a total of 14, you win regardless of what the dealer has.

In fact, one of the first things you will notice when playing this game is how the over bet, especially, acts as a hedge against most of your losing stiffs when you have a big bet out. Much of the pain of being dealt a 14, 15 or 16 vs. a dealer 10, when you’ve got a big bet on the table, is eased by collecting on the over bet prior to having to play your miserable stiff.

Another nice feature of the over/under bets is that your advantage climbs so dramatically with each true count. A blackjack player whose moderate bankroll would never allow bets above $25 would not infrequently be able to place $100 over/under bets, because his combined blackjack and over/under advantage will be 4% or more. This is virtually unheard of in any “normal” 6-deck game.

For this reason, even if you can’t afford to spread from 1-to-4 according to the strategy I’ve outlined, with your 4 unit bet placed at +3 or above, you may be able to use this spread if you wait for higher advantages for your big bet. At +3, your combined advantage is about 1%; at +4, it’s about 2 ½%; at +5, it’s more than 4%! If you get into the deeply dealt games, you will see opportunities like this regularly.

General guidelines: With a very small bankroll—say $1000—plan on spreading from $5 to $10 for most of your play according to the betting guidelines above. But look for deeply dealt games and don’t hesitate to put a $20 bet on the table if the true count goes to +5 or more. As with any other blackjack game, you’re not going to be able to make a living wage if your total bankroll is $1000. If you’ve got a $5000 bankroll, you will be able to play with quarter chips, and you will have many occasions to place the max $100 bets. But be aware of any substantial negative swings, and cut back your bets accordingly.

Future of the Over/Under Side Bet

Will the over/under game survive? Caesars Tahoe has been offering it on six tables now for more than a year, which indicates a better chance for longevity than most gimmick games. Caesars knows that card counters are attacking the game, and for the most part they’ve been welcoming them. This is because most card counters are using standard card counting systems to determine when to bet over and under, and such systems correlate poorly to the over/under effects of removal.

With the Hi-Lo card counting system, for example, you do not have an advantage on the over bet until your true count is +5 (as opposed to +3 with the Over/Under Count), and you do not have an advantage on the under bet until your true count is -8 (as opposed to -4 with the Over/Under). This means that with 4 ½ decks dealt out, the Hi-Low count would accurately recognize 3 ½ over/under betting opportunities per 100 hands. The Over/Under Count would recognize 13 ½ betting opportunities.

Most card counters using normal blackjack systems will bet far too often when the house has the advantage, simply because their blackjack systems are not designed to identify over/under opportunities. You cannot beat this game with a small spread and a normal blackjack card counting system.

So, despite this report, I suspect Caesars will continue to profit substantially from both card counters and non-counters at their over/under tables.

Player Advantage with Various Over/Under Betting Strategies in 6-Deck Games

	4 Decks Dealt	4.5 Decks Dealt	5 Decks Dealt
Flat Bet	-0.12%	+0.07%	+0.41%
1-2 Spread	+0.22%	+0.52%	+1.01%
1-4 Spread	+0.67%	+1.08%	+1.61%

Over/Under Supplement for 1, 2, 4 and 8 deck Games

Since the initial Over/Under Report was published a few months ago, one- and two-deck over/under games have appeared in Reno. Others using various numbers of decks are reportedly being offered on cruise ships and in overseas casinos. This supplement to the Over/Under Report will aid you in beating these games with any number of decks.

Assumptions

Each page of this supplement analyzes Over/Under games with a different number of decks: 1, 2, 4, 6, and 8. At the top of each page is the number of decks in play, and the assumed rules. If you are playing in a game with a different set of rules, adjust the advantages accordingly.

For example, I’ve analyzed the 2-deck game with Reno rules. The advantage at a true count of 0 is -0.7. If you find a 2-deck Vegas Strip rules game, your advantage at 0 would be -0.3. All of the figures in the advantage column should be adjusted accordingly.

For ease of use, I have analyzed your expectation with a flat bet, a 1-2 spread, and a 1-4 spread. If you are playing in a game with a different set of rules, you may approximate your advantage simply by adding or subtracting, as appropriate, the difference in your starting advantage from the “win rates” shown.

All frequency distributions assume the Over/Under Count is being used as described in the report.

Note: These frequency distributions were obtained via computer simulation with “fixed” shuffle points, so they are not symmetrical. The simulations also used a different method of rounding than the mathematical model used for the Report. For this reason, the 6-deck distribution in the supplement is different from that in the report. You may note that the 6-deck win rates in the supplement are 0.1% to 0.2% lower than in the Report. The win rates are more accurate in the supplement.

1-Deck (Reno Rules) Over/Under Count Win Rates

Count	Advantage	65%	75%	85%
-25	26.6	0	0	0.1
-24	21.5	0	0	0.1
-23	20.4	0	0	0.1
-22	19.3	0	0	0.1
-21	18.2	0	0	0.1
-20	17.1	0	0.1	0.1
-19	16.0	0	0.1	0.1
-18	14.9	0.1	0.1	0.1
-17	13.8	0.1	0.2	0.2
-16	12.6	0.1	0.2	0.3
-15	11.5	0.1	0.3	0.3
-14	10.4	0.1	0.4	0.5
-13	9.3	0.2	0.3	0.5
-12	8.2	0.4	0.6	0.7
-11	7.1	0.3	0.4	0.6
-10	6.0	0.8	1	1.2
-9	4.8	0.8	1.1	1.5
-8	3.7	1.2	1.3	1.5
-7	2.6	2.3	2.3	2.5
-6	1.5	2	2.4	2.1
-5	0.4	3.4	3.4	3.5
-4	-0.7	4.7	4.1	4.6
-3	-1.5	5.6	6.5	6.0
-2	-1.1	8.3	7.5	6.5
-1	-0.8	7.5	6.6	5.8
0	-0.4	30	28	26.8
+1	-0.1	6.3	5.6	4.9
+2	0.3	7.1	6.3	5.6
+3	1.1	4.7	5.7	5.2
+4	2.7	3.6	3.2	3.8
+5	4.2	3.3	3.3	3.4
+6	5.7	1.6	2.0	1.8
+7	7.2	1.8	2.1	2.3
+8	8.8	1	1	1.2
+9	10.3	0.8	1.1	1.4
+10	11.8	0.6	0.9	1.1
+11	13.3	0.3	0.3	0.5
+12	14.9	0.3	0.5	0.6
+13	16.4	0.2	0.3	0.4
+14	17.9	0.1	0.2	0.4
+15	19.4	0.1	0.1	0.2
+16	21.0	0.1	0.1	0.3
+17	22.5	0.1	0.1	0.2
+18	24.0	0	0.1	0.1
+19	25.5	0	0.1	0.1
+20	27.1	0	0.1	0.1
+21	28.6	0	0	0.1
+22	30.1	0	0	0.1
+23	31.6	0	0	0.1
+24	33.2	0	0	0.1
+25	34.8	0	0	0.1
Win Rate (%)
Flat Bet:		1.6	2.2	3.1
1-2 Spread:		2.5	3.1	4.1
1-4 Spread:		3.2	3.9	4.9

2-Deck (Reno Rules) Over/Under Count Win Rates

Count	Advantage	65%	75%	85%
-19	15.7	0	0	0.1
-18	14.5	0	0	0.1
-17	13.4	0	0	0.1
-16	12.3	0	0.1	0.1
-15	11.2	0	0.1	0.1
-14	10.1	0	0.1	0.2
-13	9.0	0.1	0.1	0.2
-12	7.9	0.1	0.2	0.4
-11	6.8	0.1	0.3	0.4
-10	5.7	0.2	0.4	0.6
-9	4.6	0.4	0.6	0.9
-8	3.4	0.7	0.9	1.2
-7	2.3	1.1	1.5	1.8
-6	1.2	1.5	1.8	2.0
-5	0.1	2.6	2.9	3.1
-4	-1.0	4.0	4.1	4.1
-3	-1.8	6.0	6.3	5.8
-2	-1.4	9.5	9.1	9.0
-1	-1.1	16.2	14.6	13
0	-0.7	19.6	18.5	17.7
+1	-0.4	14.6	13.2	11.8
+2	0	8.4	8.1	8.1
+3	0.9	5.3	5.6	5.3
+4	2.4	3.6	3.6	3.8
+5	3.9	2.3	2.6	2.9
+6	5.4	1.3	1.6	1.7
+7	7	1.0	1.3	1.6
+8	8.5	0.6	0.8	1
+9	10	0.3	0.5	0.8
+10	11.5	0.2	0.4	0.6
+11	13	0.1	0.2	0.3
+12	14.6	0.1	0.1	0.3
+13	16.1	0.1	0.1	0.2
+14	17.6	0	0.1	0.2
+15	19.1	0	0.1	0.1
+16	21.2	0	0.1	0.1
+17	22.7	0	0	0.1
+18	24.2	0	0	0.1
+19	25.7	0	0	0.1
Win Rate (%)
Flat bet:		0.5	0.9	1.5
1-2 spread:		1.1	1.6	2.3
1-4 spread:		1.7	2.3	3.1

4-Deck (Reno Rules) Over/Under Count Win Rates

Count	Advantage	65%	75%	85%
-14	10.4	0	0	0.1
-13	9.3	0	0	0.1
-12	8.1	0	0.1	0.1
-11	7.0	0	0.1	0.2
-10	5.9	0.1	0.1	0.3
-9	4.8	0.1	0.2	0.4
-8	3.7	0.2	0.4	0.6
-7	2.6	0.4	0.7	1.0
-6	1.5	0.7	1.0	1.4
-5	0.3	1.5	1.8	2.2
-4	-0.8	2.8	3.2	3.5
-3	-1.5	5.2	5.4	5.6
-2	-1.2	9.5	9.3	9.0
-1	-0.8	18.4	17.2	16.1
0	-0.5	26.6	23.9	22.4
+1	-0.1	16.6	15.8	14.8
+2	0.2	8.5	8.8	8.2
+3	1.1	4.6	5.1	5.1
+4	2.6	2.5	3.0	3.3
+5	4.1	1.3	1.6	2.0
+6	5.7	0.5	1.0	1.3
+7	7.2	0.3	0.6	0.8
+8	8.7	0.1	0.3	0.5
+9	10.2	0.1	0.2	0.4
+10	11.8	0	0.1	0.2
+11	13.3	0	0.1	0.1
+12	14.8	0	0	0.1
+13	16.3	0	0	0.1
+14	17.9	0	0	0.1
Win Rate (%)
Flat bet:		0	0.3	0.7
1-2 spread:		0.4	0.9	1.4
1-4 spread:		0.9	1.4	2

6-Deck (Reno Rules) Over/Under Count Win Rates

Count	Advantage	65%	75%	85%
-12	8	0	0	0.1
-11	6.9	0	0	0.1
-10	5.8	0	0.1	0.1
-9	4.7	0	0.1	0.2
-8	3.6	0.1	0.1	0.3
-7	2.5	0.1	0.3	0.6
-6	1.4	0.3	0.6	0.9
-5	0.3	0.8	1.3	1.7
-4	-0.9	1.8	2.4	2.8
-3	-1.6	4.2	4.6	5
-2	-1.2	8.8	8.9	8.9
-1	-0.9	18.7	17.9	17.1
0	-0.5	32.1	29.3	26.6
+1	-0.2	17.6	17	16.1
+2	0.2	8.3	8.4	8.4
+3	1	4	4.4	4.6
+4	2.6	1.9	2.1	2.6
+5	4.1	0.8	1.2	1.6
+6	5.6	0.3	0.7	0.9
+7	7.1	0.1	0.3	0.6
+8	8.7	0.1	0.1	0.3
+9	10.2	0	0.1	0.2
+10	11.7	0	0.1	0.1
+11	13.2	0	0	0.1
+12	14.8	0	0	0.1
Win Rate (%)
Flat bet:		-0.2	0	0.3
1-2 spread:		0.1	0.4	0.8
1-4 spread:		0.5	0.9	1.4

8-Deck (Reno Rules) Over/Under Count Win Rates

Count	Advantage	65%	75%	85%
-11	7	0	0	0.1
-10	5.9	0	0	0.1
-9	4.8	0	0	0.1
-8	3.7	0	0.1	0.2
-7	2.6	0.1	0.2	0.4
-6	1.4	0.2	0.4	0.7
-5	0.4	0.4	0.9	1.3
-4	-0.8	1.3	1.8	2.5
-3	-1.5	3.3	3.9	4.4
-2	-1.2	8.1	8.5	8.5
-1	-0.8	19.6	18.4	17.4
0	-0.5	36	33.1	30
+1	-0.1	18.4	17.6	16.8
+2	0.3	7.6	8.1	8.1
+3	1.1	3	3.7	4.2
+4	2.7	1.2	1.8	2.3
+5	4.2	0.5	0.8	1.3
+6	5.7	0.2	0.4	0.7
+7	7.2	0.1	0.2	0.4
+8	8.7	0	0.1	0.2
+9	10.3	0	0	0.1
+10	11.8	0	0	0.1
+11	13.3	0	0	0.1
Win Rate (%)
Flat bet:		-0.3	-0.1	0.2
1-2 spread:		0	0.2	0.6
1-4 spread:		0.3	0.6	1.2

On the Road to Nevada’s Nowhere Towns

by Barry Meadow
(From Blackjack Forum XX #2, Summer 2000)
© Blackjack Forum 2000

[Barry Meadow is the author of Blackjack Autumn: A True Tale of Life, Death, and Splitting Tens in Winnemucca .]

The road is long, With many a-winding turn, That leads us to who knows where, Who knows where…

Uh, I know where. I’ve been there.

Most of us have doubled down and split pairs all over Las Vegas and Reno.

Some of us have comp-hustled in Laughlin or surrendered in Lake Tahoe. But I say if you’re going to play serious blackjack in Nevada, hit the road.

That means blackjack in Tonopah and McDermitt and Ely and Jackpot and every other place you’ve seen on the map, only it was too damn far or too damn cold and why the heck would you drive a hundred miles to Lovelock when there was exactly one blackjack table in town and you’d look like a stalker trying to wong it?

I’ve been here, and there, and everywhere in Nevada. I counted cards at blackjack in every casino in the state during a two-month trip last year which consisted of me, a suitcase, and $8000. No entourage, no pals, no nothing but yours truly and one open road.

Every day, I’d record my observations on a tiny tape machine. I made more recordings in garages than a Seattle grunge band. And by the time I finished my trip, 192 casinos later, my laptop computer was in worse shape than Andy Sipowicz.

And so was I.

The name of the book is Blackjack Autumn, and if you don’t buy a copy right this minute, shame on you. The subtitle is A True Tale of Life, Death, and Splitting Tens in Winnemucca.

I should point out that the count was really, really good at the time. Once I played in a one-deck game dealt all the way to the bottom. Boy, it sure was fun taking insurance on a plus count when only two cards remained to be dealt!

Then again, there was the casino that featured a grand total of one player — me again — and seven pit employees, none of whom had anything better to do than watch me try to earn a few dollars.

At times I was the crusher, while at other times I was the crushee.

I’ve seen a blackjack dealer pull out more dead hands than a grave robber, and I’ve also seen a blackjack dealer lose so many hands that the floorman actually switched the game from two decks to one deck in an effort to change the casino’s luck (it didn’t work).

I played with plastic cards. I played a game in which a player’s two hands were dealt one face up and the other face down (don’t ask). I saw a man deal the first half of the shoe, walk to the other side of the table, and play the rest of the shoe himself.

Sometimes I was barred for card counting, including one telephone ejection at 1 a.m.. Golly, didn’t that casino manager’s mother ever tell him it was impolite to call after 9 p.m.? A card counter once told me that if you’ve never been barred, you’re doing something wrong. I guess I did something right four times.

A few tips if you’re going to make a similar trip: Bring money. Have reliable transportation. And don’t blow into town at 8 a.m., you little Carlsonite, because the blackjack game isn’t open until 4 p.m. and what are you going to do all day when there’s absolutely nothing to do?

Don’t bother showing up at 10 p.m., either, because that means you will have traveled on a two-lane Nevada blacktop at night on roads so dark that if some animal decides to cross the road in front of your vehicle, you’ll never see it until it’s draped over your hood, which is bad for your paint job.

It’s a long way from Nowheresville to South Nowheresville in Nevada, as you will soon learn. This is not a trip from Las Vegas to Henderson we’re talking about.

Let’s say, for instance, that you decide to travel to Montgomery Pass to play blackjack. Try finding it on a map. Better yet, try calling information. Chances are you’ll have no luck either way. Hint: The place is actually called the Montgomery Pass Lodge and Casino, but it’s also called Soper’s Cafe, and there’s no town anywhere nearby. If you do manage to find it, you’ll enjoy the $2 chips and the aformentioned plastic cards that the casino employees wash rather than replace. There will also be a sign advising you not to drop your cigarettes into the urinal, always a sign of a first-class establishment.

Along the way, amazing things happened to me, as they no doubt will to you should you take on this assignment. In Beatty, for instance, I realized every card counter’s dream. I won so much money at the Burro Inn — well not all that much, but pretty big for Beatty as my blackjack profits crept into the mid three-figures — that the game actually had to come to halt because if I won the next hand, there would not be enough chips to pay me. Yes, all action actually stopped as we waited for one of the bartenders to step into the casino cage to chip-run another rack of reds for my gambling pleasure.

Good thing, too, because my next hand was a blackjack.

I found an Indian casino outside of Laughlin, though I actually found quite a few more Indians near the Say When Casino in McDermitt, and I wound up helping some of them herd horses across the state highway. Found some cowboys in Mesquite, too. And highway patrolmen in more than one place, although why they found 132 miles per hour objectionable is something I’ll never quite understand — heck, there weren’t any other cars for miles and miles, for Breedlove’s sake.

I met cattle who looked meaner than Mike Tyson after a three-day drunk, and pit bosses who looked meaner still. Then again, I ran into gorgeous blackjack dealers, who made me sad that I was already engaged, although there is no evidence that any of these women would have had the slightest interest in me anyway.

Can You Win Counting Cards in Winnemucca?

Now to the important question: Can you actually win counting cards in these towns? Yes and no. You can’t win thousands of dollars, because some of these towns don’t have thousands of dollars. In Wells, the highest limit blackjack game is $25.

Forget about black chips — in plenty of places you’ll never even see green chips. If you visit the Bird Farm in Fallon, the highest amount you can play at blackjack is $5 per hand, though by brilliantly spreading from one hand of $2 to two hands of $5 I was able to carve out a $26 profit.

Nor can you blithely spread from $1 to $100 with impunity, Mr. Uston, because even the smallest places have heard of card counters, although in some places I doubt they’ve ever actually seen many of them. At Sturgeon’s in Lovelock, not only did they cut off two of the four decks, but they burned a card after every round. The blackjack game wasn’t much more attractive in Topaz Lake, or in Gardnerville, or in Jean. And I couldn’t even find the blackjack game in Amargosa Valley, which turned out to be open only on weekends.

Then again, there was Yerington. I lost there — no big news; I lost in a lot of places — but I did find one intriguing practice at Casino West. The blackjack game was four decks, with the cut card placed halfway up. When the cut card appeared, the dealer would ask the players if they wanted her to shuffle; if they said no, she’d keep dealing until only a few cards remained. Does that mean you could have gotten negative shoes shuffled away while asking the dealer to keep going on the positive shoes? Apparently so. Hmmmmm.

And sometimes the blackjack games were downright good. I had a very nice 80% penetration shoe dealt to me at the Nugget in Searchlight, and the $415 I managed to drop there was simply my way of saying thank you. And the dealt-to-the-bottom blackjack game, discovered at the Station House Casino in Tonopah, proved more than pleasant until I was asked to please leave town as quickly as possible.

Not that you’re likely to find that game any more. Game conditions in these little towns change more often than Bill Clinton’s stories, although you can generally assume bad rules. When the best blackjack game in town is four decks, dealer hits soft 17, double on 10 or 11 only, with a $50 maximum, it’s safe to assume that high-powered card counting teams will be spending their evenings elsewhere. You can’t scout these games, either — it might look a tad suspicious for a stranger to loiter around a game all day without playing, and you can’t very well table-top because there is no other open table to hop to.

There is decent action in some of these out-of-the-way places, though.

Take Wendover, for example. It’s on the Utah border, and Utah is one of only two states (Hawaii is the other) where there is no legal gambling of any kind. Naturally the Mormons who populate the state occasionally like to get down a bet or two, and so there are a couple of very large casinos there as well as some smaller ones. At the State Line Casino, I saw a guy bet two hands of $500 apiece. A warning, though: the chips in Wendover are in the weirdest colors you’ve ever seen, so be careful what you’re betting because just when you think you’re betting $17 on a hand, you just might accidentally be betting $341.

Elko gets some play, too. The owner of the Red Lion Inn owns an airline and flies in players from all over the West. Not real big players, you understand — show up with $350 in front money and you’re in, no playing obligation required — but it’s eerie to visit this small northeastern Nevada town and see dozens of players sporting junket buttons.

Comps at Casinos in Small Town Nevada

The good news is that in some of these places, if you play for anything resembling quarters, you’ll probably get comped, or least score a casino room rate. At the Saddle West in Pahrump, I had only been playing about fifteen minutes for $35 or so per hand when I asked the floorman what I needed to do to qualify for a comp; he proudly displayed the comp slip that he had already written for me, and gave me a discounted room as well. At the Tahoe Biltmore in Crystal Bay, another eager pitster offered me a room before I even had a chance to ask about it, and I had only been there about ten minutes.

Tourists in these burgs are few and far between, the action mostly confined to locals and truckers. You probably won’t find Wayne Newton in the showroom, if you can find a showroom at all. The town’s gourmet restaurant might well be the casino coffee shop. Your room will have a bed and a thin bar of soap.

But hey, it’s blackjack. Just make sure you gas up the car, because between towns is the desert, and if you run out of a gas there’s no filling station for 50 miles. It’s also probably best not to try such a trip with a 15-year-old salvage vehicle, unless you have hefty towing insurance and don’t mind waiting a week for a part to come via UPS from Las Vegas.

Another reason to double-check your automobile is that it would be highly unfortunate to find yourself alone at night on a road next to an inoperable car that’s got $20,000 in cash stuffed in the trunk. It’s bad enough that some of these places don’t have safe-deposit boxes, and you’ve got to sleep with two chairs propped up against your motel door. The last thing you want to do is break down on some bandit-filled Nevada highway, where finding some abandoned mine in which to hide your bullet-riddled body would take the average criminal maybe ten minutes.

If you take up the road warrior’s life, you don’t have to worry much about running out of casinos, since Nevada is a pretty large state. If you’ve worn out your welcome in Minden, you’ll still be welcome in Verdi.

Many places, of course, are still more than happy to bar you if you play with any skill at all, no matter how big or small they might be. After powering my way to a hefty $40 profit at Harrah’s Tahoe, for instance, I was asked to take my action elswhere, preferably to the bottom of Lake Mead. Hey, Mr. Harrah — it wasn’t me who lost millions on your New Orleans project. Where’s your sense of humor?

It’s difficult to disguise your play much in the little towns, because there is usually only one cashier’s cage, only one or two shifts, and sometimes only one or two tables. If somehow the casino is short a thousand dollars, there are very few suspects, no matter how good your act. So you’re not going to be able to camp out for a week in Battle Mountain and bury the Owl Club and the Nevada Hotel for very long.

If you want to play reds for awhile and maintain a moderate spread, chances are nothing too terrible will befall you, though your chances of getting rich are none too great, either.

Sure, the Venetian has beautiful suites and the Desert Inn has a lovely golf course, but can you stay in a historic hotel for $19.95 on a Saturday night as I did in Ely, or play a nine-hole golf course that’s only slightly more challenging than Mel’s Mini-Putt as I did in Hawthorne? OK, there are no Bellagios in Indian Springs, but exactly how many floral cornucopias do you need to see, anyway? ♠

A Parable About Winners’ Testimonials

by Arnold Snyder
(From Blackjack Forum March 1983)
© Blackjack Forum 1983

Swami Pastrami, the Great Guru of Gambling, had a dream one night that thousands of croupiers were being strangled by their bow ties while drowning in 100-gallon vats of tomato soup. The following morning he sent an epistle to his one million followers: “I have had a vision! Send me just $10 and I shall share with you the secret of acquiring unlimited wealth!”

All one million of the Swami’s followers sent him the money he’d requested, making the Swami a very rich holy man. The Swami sent each of his disciples his interpretation of his strange dream: “Find a croupier with a crooked bow tie and bet one hundred dollars on red. Parlay your winnings until you are rich!”

Unbeknownst to Swami Pastrami and his followers, the Swami had misinterpreted his dream, which was not, as the Swami had presumed, of divine origin, but had actually resulted from the Swami’s over-indulgence in jalapeno peppers at dinner the previous evening.

Within the next month, all one million of the Swami’s followers had found croupiers with crooked bow ties and placed $100 bets on red. Since the chance of winning this bet is only 18 out of 38, it came to pass that 526,316 players lost their $100, while 473,684 players won. Those who had lost left dejectedly, but those who had won praised the Swami and soon all the winners had placed $200 bets on red.

As expected, 249,307 lost their money and left the tables. But 224,377 players won, and now excitedly bet $400 on the red square. This time 118,093 lost, while 106,284 won. These players, with $800 each, were now firm believers in the Swami’s systems, and so were the casinos! Since the Swami’s followers had begun playing, the casinos had already won a combined total of almost $15 million just from the Swami’s disciples! Word spread from casino to casino, and soon there was a standing rule that any follower of Swami Pastrami would be covered for any bet of any size at the roulette tables. There wasn’t a croupier in Las Vegas with a straight bow tie.

The 106,284 remaining followers each placed $800 bets on red. In no time at all, the Swami’s followers were reduced to 50,345, each with $1600. After another bet, the remaining players were cut to 23,848, each with $3,200. Then they became 11,296 players, each with $6,400. One bet later, there were only 5,351 players left, with $12,800 each.

After yet another parlay, there remained 2,535 players, each with $25,600. These players were now worshipping the Swami publicly. One bet later, however, their numbers were reduced to 1201 followers, each with $51,200. The casino had now profited more than $38 million from the Swami’s followers.

The Swami, meanwhile, had a new ad campaign going, which was based on honest facts: “More than 400,000 of my followers doubled their money the first day!” More than 1200 of these dedicated disciples have watched their $100 bankrolls grow to more than $50,000!”

There is a moral to this story.

Remember that literally millions of gambling systems and books on how to win have been sold over the years. The fact that an individual has made money using some inexplicable system is not proof that the system is valid. Beware of system sellers who use “testimonials” from successful players in their advertising. You’ll find this form of advertising popular for craps, roulette, baccarat, keno and some blackjack systems. Although the testimonials may be real, the short run experiences of individual players are meaningless.

Incidentally, if you are wondering what ultimately happened to the Swami’s 1201 successful disciples, who had $51,200 each, eight parlays later only three followers were left, each with more than $13 million. By this time, the casinos had profited more than $60 million from the ex-disciples.

Noting the continually-shrinking ranks of the Swami’s original followers, the three remaining disciples wised up and decided not to place any more bets on the wheel. Instead, each disciple paid Swami Pastrami $1 million for the honor of opening a Pastrami Roulette Franchise in his own neighborhood. They all made their second $13 million twice as fast as their first. ♠

Shuffle Bored, Anyone?

by Arnold Snyder
(From Card Player, March 1993)
© Arnold Snyder1993

The non-random shuffle gurus are making the rounds again. Or, at least, there are some new nonrandom shuffle gurus making the rounds. Every few years for the past decade, some self-proclaimed genius starts hustling a blackjack system based on the fact that casino shuffles do not distribute the cards randomly. For a few hundred bucks, one of these brilliant system developers will sell you the inside scoop on how to play blackjack by following the “trends,” “clumps,” and “biases.”

This all started ten years ago with the TARGET system, an invention of Eddie Olson, later hawked by Jerry Patterson. Many variations have arisen since then, but the theory and playing methodology never really changes much.

Here’s how Swami Nonrandami’s logic goes:

First, it’s necessary to acknowledge that casino-style shuffles are less than perfect, and that the cards are not randomly distributed by these shuffles. No problem, since anyone who has played any length of time at casino blackjack tables can see that sloppy shuffles are easy to find. When new decks are brought in, it’s not unusual to see occasional cards being dealt in consecutive new-deck order. So we know the shuffles are imperfect.

Next, you must accept the fact that these nonrandom shuffles are affecting the decisions on the hands dealt. No problem again. If you happen to see a dealer hit his fourteen with a six of spades, right after you doubled down on your eleven vs. his four up — you having caught the spade five — then you will be a believer. Yes! Yes! Those nonrandom clumps are killing me!

Now, what if you had a system designed to play those clumpy games? A system that made rational assumptions about hitting and standing based on the severity of the clumps? Yes! Yes!

Finally, a blackjack system that takes into account the kinds of weird stuff we actually see in the casinos. It’s not a system based on some mathematician’s analysis of some computer programmer’s simulated billion hands of play. This is a reality-based system, and that’s the only kind of system that works in the real world.

Card counters are out there talking about advantages of 1%, and they don’t even realize that the casinos sometimes have a 10% advantage over them, based on the nonrandom shuffles. What’s worse is that the same counters don’t realize that they can get a 10% advantage over the casinos, courtesy of the same lousy shuffles.

Etc., etc., etc., etc., etc., etc. . . .

A lot of players buy this baloney, and to be honest, it sounds very legit. There’s only one thing I don’t like about Swami Nonrandomi’s “logic,” and that is that it cannot be demonstrated by computer simulation.

John Imming’s Real World Casino (RWC) software allows programmable, nonrandom, casino-style shuffles. The deck(s) begin in regulation new deck order, and the shuffle routines simulate actual riffles, strips, cuts and washes, as fine or as clumpy as you decide, even utilizing casino-style breaks into multiple shuffling segments if you so desire.

Here’s what I’ve found with the RWC software so far:

The biggest effect on the player’s expectation I could find comes from no shuffling whatsoever. Ironically, this is a player advantage, not a house advantage. I’ve tried Imming’s software with 1, 2, 4, 6 and 8 deck games, with both lay & pay, and pick & pay, dealing styles, and the player advantage rises by .70%-.75% if playing one-on-one with the dealer, regardless of the number of decks in play or the pick up style. Somehow, the play of the hands puts the cards into an order that favors the player.

Both Stanford Wong and John Gwynn had independently discovered this years earlier. Wong, in fact, ran a computer analysis to determine in what way the play of the hands ordered the discards differently from random, and he discovered that in the discard pile high cards do tend to clump with high cards, and low cards with low cards. We don’t know why this favors the player, but it does.

As multiple players are added to the table, this no-shuffle player advantage diminishes. For some reason, the first base side of the table retains the advantage, but the third base side loses it and then some.

Once you start adding any type of shuffle at all to the game, the (dis)advantages diminish, until the real world shuffle results are indistinguishable from random-number-generated shuffle results. The biggest effect I could find in a simulated casino game, utilizing what I figured to be the sloppiest shuffle you might realistically expect to find, was a couple tenths of a percent more or less than the normal basic strategy expectation.

My attempts at creating a sloppy shuffle which would have a greater effect than this were unsuccessful, even though the RWC software allows unlimited variations on lousy, inadequate shuffles.

So, where is this monstrous effect that Swami Nonrandami is crying about? I just don’t buy the explanation that it happens in a casino, but not in a computer. Why not? New deck order is new deck order, and nonrandom sloppiness is nonrandom sloppiness. There’s nothing magical about a lousy, lopsided riffle that a computer can’t simulate.

But there is one factor all the nonrandom shuffle gurus have in common. They all say: “Oh, by the way, you can’t simulate this effect on a computer.” Yet they spout all kinds of precise percentages, based on their “personal studies.”

I say, “Baloney.” Computers may not be able simulate everything under the sun, but card games are one of the things computers are very good at simulating, especially if what you’re looking for is the player’s expectation vs. a fixed house strategy. So take a hike, Swami. I don’t believe in gambling systems based on faith. If you can’t do the math, hit the path.

If you want information on legitimate professional gambling techniques for exploiting non-random shuffles in blackjack, see my book Blackbelt in Blackjack to get started. If after reading that, you decide you’re up to the challenge of actually learning to win with shuffle tracking, see The Blackjack Shuffle Tracker’s Cookbook: How Players Win (And Why They Lose) with Shuffle Tracking.

Send items of faith, hope and especially charity, to the Bishop.  ♠

A Review of E. Clifton Davis’ NBJ System

by Arnold Snyder (with commentary by the Boardwalker)
(From Blackjack Forum XIII #3, September 1993)
© 1993 Blackjack Forum

[Note from A.S.–WARNING! WARNING! If you are considering buying the NBJ system, you are considering investing money in a Martingale betting system! STOP! Consult with a mathematician (a real one) immediately! Emergency mathematicians are now on duty to take your call!]

1991: Enter E. Clifton Davis and NBJ (“New Blackjack”), a dressed-up martingale betting progression system for casino blackjack. The system is published by Jerry Patterson Enterprises.

Davis has been an occasional contributor to Eddie Olson’s Blackjack Confidential newsletter. Olsen is the inventor of the TARGET system, one of the first widely promoted “streak” or “trend” blackjack systems — also sold by Patterson.

I wasted a lot of ink trashing the trendies in the past. Ed Thorp, Ken Uston, Stanford Wong, Julian Braun, Peter Griffin, and Mason Malmuth have all gone on record stating their shared belief that the TARGET blackjack theories are worthless. Every independent computer simulation of non-random casino-style shuffles that I’ve seen refutes the TARGET theories about the effects of poor shuffles.

The Facts About Card Clumping

Here are the facts. When cards are dealt to players who all play perfect basic strategy, the discard stack, if dealt again without being disturbed by a shuffle, would be in an order highly favorable to players. John Gwynn and Stanford Wong independently determined that the play of the hands at blackjack puts the cards into a player-favorable order. Players can confirm this themselves with John Imming’s UBE software.

Low cards and high cards do get clumped together during the play of blackjack hands, and this clumping would be worth about .75% to the basic strategy player if the cards were not put through a shuffle before the next deal. (This is determined by running simulations with no shuffle, just picking up the discards, cutting, and resuming play.)

It’s even possible for brand new decks to favor the first base side of the table by a few tenths of a percent, and similarly hurt the third base side of the table, if you put the new decks through a weak enough shuffle (weaker than any of the moderators at BJF have seen in any casino in the world–one riffle, no cuts, no strips on one-deck).

But it does not appear possible to create a shuffle that has the huge advantages/disadvantages (+/- 5% to 20%) described by the non-random shuffle gurus. In any case, casinos do indeed put the cards through a more thorough shuffle before dealing discards or new decks, and the shuffle is sufficient in every casinos we’ve seen to eliminate this player-favorable effect.

Plus, I have personally tested Martingale systems, like those advised by Davis, vs. extremely sloppy shuffles using Imming’s UBE software. There is simply no effect, regardless of how random, non-random, or “clumpy” the cards are.

It was ten years ago — September, 1983 — when my first TARGET article appeared in Blackjack Forum. Since then, Blackjack Forum has picked up a lot of new readers, many of whom did not follow the TARGET controversy as it developed.

So, here comes E. Clifton Davis with NBJ, as well as his new improved version of NBJ, which he calls WCB (“World Class Blackjack”). I’m regularly getting news clippings of ads for “New Blackjack” introductory seminars from all over the country. Players are sending me direct mail advertisements for NBJ they’ve received from Jerry Patterson.

I’ve even got a complete copy of the 119-page “New Blackjack” Home Study Manual, written by E. Clifton Davis, provided by one of Davis’ students. I’ve got an audiotape of a 45-minute interview with Davis by Jerry Patterson, in which Davis expounds on his theories. This promotional tape is accompanied by a 9-page advertisement for Davis’ “World Class Blackjack” system.

As of the writing of this article, “New Blackjack” sells for $445 mail order. WCB is now selling for $500, though Patterson’s letter states that this is an “introductory offer” to former NBJ students, and that the price of WCB will probably be going up to $1000.

Reports on NBJ from Players

This is one letter I received from an NBJ player:

[Regarding] E. Clifton Davis’ NBJ course… I asked for (and got) my money back because I didn’t play it to the 5%-15% advantage that the system touts. I did, however, win with it. My records reveal that I played it to a 1.96% advantage during three weeks last spring — certainly not a large enough trial to compel anyone to change religions (so to speak), but I wasn’t exactly disgusted with the outcome either, since seldom do I perform that well using a conventional counting system.

If I may be allowed to free-associate (don’t a lot of Californians do that???), my thoughts on NBJ are as follows:

I went to the seminar as advertised, and the following assertions were really made: That currently, the typical house advantage against a basic-strategy player is about 20%. No, not a hold percentage of 20% (which is believable), but a house advantage of 20%. In addition, they claim that because of like-card clumping, the current dealer bust percentage is about 12%…

Furthermore, Davis pledges that to keep the casinos from making any wholesale changes in their procedures resulting from any of the material in this course, under no circumstances will he sell his course to more than two percent of the blackjack-playing public.

Let’s do the math, shall we? 30 million blackjack players times 2% times $445 each (the cost of the course) equals $267 million. You gotta admire the man’s character for placing those kinds of economic restrictions on himself.

Finally, Davis asserts that 92% of his students win, compared to 1% for card counters, and that one fellow in Minnesota won 41 straight sessions with this system…

As for the course itself: I think that Davis’ advice on using betting progressions is ludicrous. To me, it’s like tracking roulette results, determining a pattern, structuring a progressive betting system to match that pattern, and hoping that you’re not too wrong if that pattern collapses. In his course, Davis has the student ascertain a “game type,” which he says sustains itself from one shoe to the next, and then use a betting progression to exploit it. Maybe it was just me, but for me the game types just didn’t remain the same from one shoe to the next.

I also think his strategy charts need a little refinement in places. The basis of NBJ, as Davis says, is non-random cards (due to insufficient shuffling, particularly in shoe games). As a result, sometimes you’re instructed to violate basic strategy plays, depending on whether high or low cards “are running.”

One such play is to split 6’s when the dealer has an ace up and “tens are running.” When I called Davis myself to ask him about this, his response was that the player’s objective in this situation is to go for an overall push by winning one of the two hands. Whaaaat?

But I also believe that there are some viable concepts in his system, particularly with respect to the notion that cards are not sufficiently shuffled so as to be randomly distributed. Although there was one occasion where I didn’t split a pair of aces that I was dealt (because “low cards were running”), then got two tens, there seemed to be quite a few occasions where his methods could be used to determine whether the dealer had a stiff hand with that ten up, and/or what my next hit card would be based on the values of the previous four or five cards.

As is the case with first-basing or playing tells (in blackjack or poker), you don’t have to be right 100% of the time to make your system work. And if I can consistently use this non-random-card business to change an average of two losing hands into two winning hands each hour (over and above the number of winning hands turned into losers by using this method), I’ll be changing careers real soon.

Conclusion: I’m still on the fence. I’ll be experimenting heavily later this year to see if non-random card analyses can be used in conjunction with conventional counting to eke out another percent or so advantage. I’ll let you know the results, both in a simulation environment, and in the casinos.

I think that NBJ might warrant a comment from you and/or other blackjack authorities since, according to Davis, his graduates now number over 1,000. Since one could argue that there is some subjectivity involved when making NBJ playing decisions, do you think that it’s conceivably possible for a computer jock somewhere to simulate this method?

Possible, yes. Probable, no. Many reputable programmers have lost interest in testing betting systems because it has been proven over and over again that such systems do nothing.

Again, I have personally tested Martingale systems, like those advised by Davis, vs. extremely sloppy shuffles using Imming’s UBE software. To repeat: there is simply no effect, regardless of how random, non-random, or “clumpy” the cards are.

The author of the above letter seems to be a thoughtful and intelligent person. His questions about NBJ reflect a genuine interest in whether or not some aspect of Davis’ approach might be valid, even if there are many aspects of the system which he finds hard to swallow.

Blackjack Forum’s “Atlantic City Update” columnist—who writes under the nom de plume The Boardwalker—was less kind in his remarks about NBJ. The Boardwalker never bought the system; he just attended one of the $10 NBJ introductory seminars that appeared in his area. This is his report:

NBJ = (FOH + BBS)^CST

The newspaper ad read like a carnival pitchman’s bark: E. Clifton Davis’ New BlackJack is sweeping the country! Why counting no longer works! Triple your bankroll! Win more hands than the dealer! How casinos make you lose! Win 75% of your double downs! Why basic strategy makes you lose!

Being a doubting Thomas for the ’90s, you couldn’t have kept me out of this seminar for an RFB comp. So, I plunked down a sawbuck and grabbed a chair right up front.

Arriving early, I scanned the four page NBJ sales leaflet, which had 10 playing tips from Davis on the front. Tip #7 said to always insure a natural against a dealer’s ace. Knowing this is a 4% basic strategy error, my antenna went into high gain. Tip #9 said to treat a dealer’s deuce like a ten. Not even if you held a gun to my head. Well, maybe for that.

Inside were more gems such as: win 80 units per hour; learn a winning system in 20 minutes; win 54% of your 15’s and 16’s; win 67% of your insurance bets. And all this without having to count. Could I plan to retire on just that last one alone?

A few of the 100+ blackjack curious in attendance and I had a chance for a friendly chat before the main event. I played the part of blackjack moron and let them spill their guts. If their knowledge was any kind of a representative sample, then in my opinion, many couldn’t recite proper basic strategy if asked and those who said they knew how to count cards wouldn’t know a ruin formula if it jumped off the page and bit them on the nose! Cherries, ripe for picking, if you ask me.

Enter the pitchman, Michael Simpson, a full time investment banker and part-time NBJ player/salesman. He began with the words of a confidence man, saying, “You won’t hear the truth anywhere else.” Opening remarks included highly questionable statements like, “Basic strategy used to work and so did card counting.”

Simpson claimed that basic strategy players win 40% of the hands and the dealer wins 60%, therefore, the casino enjoys a 20% advantage. When asked about the effect of doubles and naturals, he muttered some mumbo jumbo that made no relative sense and changed the subject.

Simpson said that NBJ has nothing to do with counting cards, yet in the same breath noted that while a 9 is of little value to card counters, it is a valuable card in the NBJ strategy. Go figure.

In a continuing ramble, he submitted that NBJ is based on card clumping and that after cards are recognized as being properly clumped, they can be predicted, giving the NBJ player a significant advantage over the house.

Simpson went so far as to say dealers intentionally perform high-low stacks while picking up completed hands, then shuffle in such a manner as to complete the stack. (Kudos to Steve Forte for educating me on these techniques in his Gamblers’ Protection videos.)

Besides the fact that I have never, ever, witnessed this in my 1000-plus hours of casino time over the years [note from A.S.–now 15,000 hours and still playing], there is absolutely no legal way the casino could use this to their advantage. Yet Simpson claims NBJ players can recognize this happening and can use it to predict hit and hole cards with 50-75% accuracy. Not likely, in my opinion.

I asked Simpson if NBJ has an insurance strategy and he replied, “Yes, we do insure and we do it very well.”

I asked Simpson how NBJ players assess whether or not they are playing properly and he replied, “If we won, then we made the correct play.”

Finally, I asked how many hours he had used NBJ in actual casino play and how much he was ahead. Simpson said he would rather not say in a public place, but added that he has played recreationally for 2 years and was ahead multiple thousands of dollars. Modest, but not too much so and also still very much in the short run, wouldn’t you say?

On my way out the door, I was given an NBJ newsletter for prospective pigeons. It says 40 units per hour is often earned in a good “type 1” game, whatever the heck that is. NBJ recommends players start small and gradually build up a bankroll, and gives these players specific low risk procedures to follow for a while. I guess once you’ve started making your fortune, you can advance to higher risk play.

 One comment that got my water boiling was that NBJ players don’t stand out like card counters because they don’t cheat. That’s right. NBJ players stand out at the casino cash advance machines. Sorry, I added that. Couldn’t resist.

Also included in the newsletter is a 2 1/2 page testimonial to E. Clifton Davis by Jerry Patterson, which attempts to put Davis on a higher pedestal than Thorp, Griffin, Wong, Uston, et al. But let’s face it. While I’ve heard Patterson is a pretty nice guy, his credibility in the blackjack world has been questionable since the early eighties.

The remaining pages of the newsletter contain a table of contents from the NBJ manual, which includes such amusing topics as “Telltale Signs of Clumping,” “Testing the Water,” “Negative Betting Progressions,” and “Insuring for Less.” Tell me something, if you have a known positive insurance expectation, why would you do it for less? Better yet, would you spend $445 on the NBJ system to find out? I think not. The friendly folks at NBJ will also sell you a different “World Class” system for another $500.

The NBJ people did seem to be genuinely friendly because I called and talked to a couple of them. I spoke with Marv, who considers himself a professional blackjack player, and Suzanne, who is a part time NBJ player and has contracted with Davis to sell the NBJ systems in my area.

Marv sounded like a really nice chap over the phone and was more than willing to tell me about how important 1st and 3rd bases are to NBJ players and that he can control the table from 3rd base, presumably by taking the dealer’s bust card or sticking him with one.

He felt positive that he could guess the dealer’s hole card at the rate of about 80% the other night using NBJ techniques. But Marv was quick to add that you don’t want a whole table full of NBJ players because they take all the good cards from each other! I’m sure the casinos would say the more the merrier.

Likewise, Suzanne raved about how well the system has worked for her. I asked her if Davis supplied mathematical proof for his theories and for some reason she began telling me about his credentials.

She did contradict Marv, however, by telling me the dealer will break more often with more NBJ players at the table and that with NBJ, one can predict hole cards up to 90% of the time.

Marv and Suzanne did agree on two things, though. One of the keys to playing NBJ is using the “tens ratio,” which you derive by observing the number of tens on the table. But hey, that’s not counting because NBJ players don’t cheat.

Also, I just had to ask them both if NBJ included anything on ruin probabilities or anything like that. Both were quick to respond that NBJ uses a 12-unit stop-loss money management strategy. Marv even goes one better. If he loses 3 units, he walks.

I could go on for at least another few pages, but I’m sure the readers of Blackjack Forum have the general idea. However, in case you’re curious and haven’t tried to figure it out yet, the title of my report means New BlackJack is Full Of Holes and Basically BullSh*t to the Casinos Say Thanks power!

---

What’s Taught at the NBJ Seminar

Okay, so the Boardwalker wasn’t exactly enthralled by the NBJ seminar he attended. I have since spent a considerable amount of time reading and analyzing the NBJ Home Study Manual because NBJ is not a system which will only appeal to dummies (as one might assume from the Boardwalker’s report).

NBJ is a complex system, which requires that the player first ascertain which one of six “game types” he is playing in. The patterns of wins and losses determine the game type. The player’s betting strategy will then be one of many recomended Martingale, or sometimes reverse-Martingale, betting progressions, depending on the game type. The hands are played according to a variable strategy, depending on how the player reads the clumping effects of the high and low cards. I couldn’t even begin to explain the details of the system in a short review, but that’s not necessary. What is most intriguing about Davis’ NBJ system are the theories behind it.

Regarding Davis’ Tip #9, from the hand-out which the Boardwalker received—“Treat a dealer two up like a ten”—I would assume Davis is advising less splitting and doubling when the dealer shows a two up, and more hitting on stiffs.

Most of Davis’ deviations from basic strategy, which seem weird at first perusal, would fall into the category of bankroll conservative. The other two deviations on this tip sheet—not splitting 8s vs. 9 or 10, and always insuring blackjacks—would both tend to reduce fluctuations even if they are technically incorrect plays. The fact that Davis includes advice to never split tens (Tip #8) indicates that he may be assuming a low level of expertise in many of the players who take his course.

Tip #10—When in doubt — hit—would also strike most knowledgeable players as very strange advice. If we consider, however, that the most consistent error of poor players is failing to hit stiffs vs. dealer high cards, Tip #10 might not be such bad advice for a lot of neophytes. Nobody is going to lose his shirt by always insuring his blackjacks. Some pros do this religiously as a form of low cost camouflage.

The Boardwalker is correct that this is an expensive error—but it does not come up frequently enough to hurt anyone significantly in the long run. It is also a fact that always insuring your blackjacks will reduce your bankroll fluctuations. Again, not significantly because of the infrequency of the hand, but combined with all of the other strategy deviations Davis recommends, an NBJ player would experience significantly less severe fluctuation than a basic strategy player.

The most effective technique Davis uses to reduce fluctuations is his conservative double down strategy. NBJ never advises doubling down on any player hand vs. any dealer upcard as basic strategy. Doubling down is only advised after the player predicts both the dealer’s likely hole card and the likely hit card the player would receive, depending on the game type, whether or not high cards or low cards are running, etc.

Davis really attacks the double down basic strategy, and all of the so called “basic strategy experts,”—and he likes to put “experts” in quotes. He says basic strategy and card counting fail because the cards aren’t random. NBJ players, au contraire, play on a “higher level.” They exploit the win/loss trends by card predicting. According to Davis, this is why NBJ players win such a high percentage of their double downs.

Actually, NBJ players should win a greater percentage of their double downs—but not for the reasons Davis states.

Any blackjack “expert” knows that a player who never doubles down (assuming he follows the optimal hit/stand strategy) will win more hands than a player who follows correct double down basic strategy. This is elementary. Any time you double down vs. a dealer high card (7-A), you relinquish the right to take another hit should your double down card make you stiff. Doubling down is essentially agreeing to win a smaller number of hands in order to win more money in the long run due to more action on hands that are worth the risk.

Doubling down less often, as Davis advises, would not only result in NBJ players winning a greater proportion of their hands, as he says they do, but they would also experience less volatile fluctuations to their bankrolls than if they followed double down basic strategy. (Hey, if you’re going to sell a Martingale betting progression to the general public as a wise investment, you’ve got to take what steps you can to reduce fluctuations!)

Basic strategy was not devised to reduce fluctuations, but to optimize the player’s expectation in the long run. Doubling down less often is not really a wise long run strategy for a player who wants to beat the house. If you do not take every opportunity to risk more money on favorable hands, you will not beat this game. If your bankroll cannot afford the risks associated with doubling down, you probably shouldn’t be playing blackjack. Watch out! Wake up!

The NBJ variable strategy—as opposed to basic strategy—is really not that bad. True, there is some weird advice that I’ll be damned if I can figure out. (There is a distinct possibility that E. Clifton Davis may actually be from another planet…)

But basic strategy is advised for most hands, with variable strategies allowed for the more borderline hands (depending on how the cards are “running”). The variable (and therefore less frequent) pair splits and double downs are the major differences from traditional basic strategy.

I tried playing the NBJ strategy against one of my computer practice programs (Blackjack—Your Key to Winning Play)—which, of course, must be a sacrilege of some sort. I’m not claiming I actually learned this system… I get the feeling nobody could ever quite “learn” it, since decisions are “intuition” based.

Attempting to employ the system, with the book in my lap, I discovered that—intuitively—I almost never doubled down vs. dealer high cards. Consider: if highs are “running,” there is too much risk that the dealer’s hole card will be high, giving him a pat hand. If lows are “running,” there is too much risk that I’ll make myself stiff. About the only time it ever seemed safe to double down was when the dealer showed a low card, and highs were “running.”

Double downs vs. dealer low cards are, in fact, the most profitable double down plays. Double downs vs. dealer high cards (even when correct) are the riskiest plays. This is not a new revelation, but a simple fact. Davis somehow never mentions this. I’m sure NBJ players do, in fact, win more of their double downs than basic strategy players. But I do not believe NBJ players win more money on their double downs, as Davis’ analysis—based on his private research—finds.

I consider it nonsense that an NBJ player could predict that highs or lows would continue “running,” but if an NBJ player uses his estimation of such factors to make double and split decisions, he will play a more bankroll conservative strategy than a basic strategy player or a card counter, he will win more hands, and he will win more of his doubled bets. I can almost see the above words on the cover of Davis’ next NBJ newsletter. “Arnold Snyder States That NBJ Players Win More Hands!”

Actually, Davis’ theories and explanations are bunk, but he has developed a fairly intelligent style of play for the betting progression system he is selling.

If you are going to play a betting progression system, and especially a Martingale progression like the ones Davis touts, you would want a system designed to win the greatest number of hands. To win a Martingale progression, it only takes one win. A large bet on the table is not indicative of a large advantage, as with card counting, but of a previous series of losses. It would be foolish to double your bets in risky situations with a Martingale strategy. You want to win your series so you can quickly revert to a single unit bet again.

Davis acknowledges this quite blatantly on page 93 of his Home Study Manual, when he explains the “most important reason” for always insuring your blackjacks: “…we aren’t just risking one hand. We are usually risking an entire progression…”

His logic is flawless. If you always take even money for your blackjack, instead of playing it out, you will win your series with certainty. Why play out the hand, risk pushing the dealer, and then risk losing the series on the next hand? A card counter can pull his bet back after pushing a dealer blackjack, but a Martingale man has to win as many series as possible, and abandon as few as possible. Davis’ Martingale strategy is very conservative, since he advises abandoning a series, and reverting to a one-unit bet, after only three consecutive losses. On a coin flip, a player would win 87.5 series out of a 100 with this betting progression. On his 12.5 abandoned series, his average loss would be seven times greater than his average win, and in the long run, he would break even.

Unfortunately, unlike a coin flip experiment, multi-deck casino blackjack is less than a 50-50 proposition without a betting strategy that is based on an intelligent analysis of the mathematical advantage.

Davis does not advise players to use NBJ in single-deck games, nor would I. My reason for not advising it is different from what he says. In single-deck games, a player who bases his strategy on “runs,” with the assumption that the “run” will continue, will be playing at total odds with count logic. My count goes down when I see a run of tens on the table, and such an occurrence would lead me to play as if I were less likely to be dealt a ten.

With six or eight decks, however, a current “run” of high cards or low cards will have a relatively minor effect on the hand probabilities. Davis is still at odds with count logic, but the count is less volatile in shoe games.

For many casual players who do not have the dedication to learn a card counting system, and who do not seriously entertain fantasies about professional play, NBJ is not that bad of a strategy. It might save some players from making some of the more expensive hunch plays, and could also discourage overbetting in players who have a tendency to “steam.” I feel sorry for players, however, who believe that NBJ will turn them into blackjack pros.

The system should appeal to those who prefer to embrace all of the “common sense” myths about blackjack which we know to be baseless. Insuring blackjacks. The third base player controlling the table. Bad players affecting other players’ expectations. Dealer hot streaks and cold streaks. Davis does not really explain the real logic of his system, as he does not admit that NBJ is just a complex progression system. He acts as if he’s discovered some magic blackjack secret based on the non-random shuffle.

Some of his explanations are a howl. On page 15 of his manual he insists that one of the reasons computers cannot be used to analyze casino blackjack is because you cannot program in “the humidity,” and also, there are “…certain player types who can change the odds of the entire game.” He has no shame about spouting such nonsense.

Many players who use NBJ might feel that they are winning more than they are losing. One reason is that they will win more hands. Another is that they will, in fact, win so many more of their betting series than they will lose, that it will feel as if they are winning much more. In fact, some will win, and some will lose. More will lose in the long run.In short runs of play, the positive and negative fluctuations will be pretty wild.

NBJ is not a cut and dried system. The NBJ player is encouraged to “educate” his “intuition,” and to play according to it. If you predict a hole card or hit card incorrectly, it’s not necessarily the system’s fault, it could be your fault for not reading the cards correctly. There’s a chapter on “enhanced card reading,” which is to card counting what numerology is to arithmetic. In my opinion, the Manual leaves a lot of questions unanswered.

Regarding Jerry Patterson’s TARGET system, Davis admits that TARGET and NBJ are “compatible” and “compliment each other.” There’s a chapter titled “Target and NBJ—The Perfect Marriage.” Now there’s a match made in heaven…

NBJ begins with the TARGET premise that the non-random shuffles make the game predictable and exploitable without card counting. On page 41 of his Manual, Davis provides “proof” that casino games are not random. Simple observation, he informs us, tells us that there are “good games and bad games.”

If the shuffles were random, he reasons, “…all games would be the same.” Using this same reasoning, if I were to flip a coin one hundred times in succession, always betting on heads, and then if I were to do this series of one hundred flips again and again, always betting on heads, I should never experience “good games and bad games” with an honest (random) coin. In fact, multi-deck games are not always well-shuffled, and certainly aren’t randomized. I have no argument with that. Professional players do exploit these games, but only with mathematically justifiable methods.

Basic strategy and card counting have been computer tested extensively in poorly shuffled games. Numerous articles have been published in Blackjack Forum in the past decade reporting on these findings. Davis simply ignores all of the literature on this subject.

One amusing note: throughout his Home Study Manual, Davis concludes many of his analyses with the words: “It’s a thinking man’s game!”

On page 56, Davis teaches us that traditional basic strategy assumes that the dealer’s hole card is a ten. However, he reasons, since only 30.8% of the cards in a deck are tens, he can improve on basic strategy simply by coming up with a hole card prediction that is correct more than 30.8% of the time! (Run that by me again…?)

Most serious players realize that basic strategy considers the distribution of all of the cards in the deck(s). No assumptions are made about the dealer’s hole card, other than its proportionate likelihood of being any one of the available cards.

In any case, even if we accept the betting progressions as bunk, and the strategy deviations as simply designed to win more betting series, and all of Davis’ theories about basic strategy and card counting as tongue-in-cheek humor, is there any possibility that Davis is suggesting anything of value to the player? Is it conceivable that a player might profit from playing his hands differently according to how the high cards and low cards are “running?”

To be honest, I’ve never seen a computer simulation of casino blackjack in which the player made strategy decisions by predicting that the short run pattern of the cards would continue. Personally, I tend to doubt that there would be any value to such a strategy. Disregarding the humidity factor, it seems to me that a computer-simulated casino-style non-random shuffle would suffice for the test.

It would also be necessary to define much more specifically than Davis does exactly how to determine the type of game, and what quantities and proportions of high, low and middle cards determine when something is “running” or stops “running.” Computers don’t have a lot of intuition. Some of Davis’ theories could be computer tested, but his NBJ system, as it is presented, could not. There is just too much guesswork.

If, despite my remarks, you believe Davis’ theories are worth investigating, then I’d be interested in hearing about your personal experiences with his methods. If anyone has tested card “running” strategies via computer simulation, we would be interested in your findings. I did talk with one other Blackjack Forum subscriber about NBJ, and he liked the system, though he acknowledged very limited casino experience with it.

From what I’ve seen, Davis is simply selling a betting progression system, and calling it the road to riches. Same old baloney. ♠