This paper was posted by Michael Hall on rec.gambling.blackjack in 1990 and later was copied to bjmath.com. Alas bjmath.com has disappeared and with it this valuable paper, one of the few documents on the subject. I am trying to locate Mike through mutual friends for permission to post it permanently. Unfortunately, he also disappeared from the scene in 2001. As I look, I'll keep the paper here. I can't imagine that he would mind as it has been posted publicly for nearly 20 years.

Copyright 1990, 1991, Michael R. Hall

INDEX

=====

INTRODUCTION

THEORY AND EVIDENCE IN SUPPORT OF SHUFFLE-TRACKING

- The Theory of Shuffle-Tracking

- A Simple Shuffle-Tracking Strategy

- Empirical Results in Support of Shuffle-Tracking

THE PRAGMATICS OF SHUFFLE-TRACKING

- How to Track Without Going Crazy

- Fine Points of Shuffle-Tracking

- How to Avoid Casino Countermeasures

REAL WORLD SHUFFLE-TRACKING STRATEGIES

- The Criss-Cross Zone Shuffle

- The Random Pick Order Six Zone Shuffle

- The (Dreaded) Stutter Shuffle

- The Stutter Plus Shuffle

OTHER SOURCES OF SHUFFLE-TRACKING INFORMATION

APPENDIX I: Glossary of Terms

APPENDIX II: Hand Trial Shuffle-Tracking Empirical Results

APPENDIX III: Computer Trial Shuffle-Tracking Empirical Results

INTRODUCTION

============

Shuffle-tracking is a mathematically-based approach, just like card

counting. In fact, shuffle-tracking is based on card counting.

The premise of shuffle-tracking is that shuffles are nonrandom -

by this I mean that the location of cards after the shuffle is to

some degree predictable. Counting is necessary to have some idea

of the favorability of different regions of the played cards, so

that these regions may be tracked through the shuffle.

A common shuffle used by casinos is the "zone shuffle". Here, the
cards

are broken into piles, and then the shuffling is only performed between

the piles. Thus, even with the uncertainty in pick sizes and riffs, a

particular card has a zero percent probability of being in certain

portions (most of) the shuffled pile, and a high probability of being in

one or two particular portions of the shuffled pile. Casinos do not

use more thorough shuffles, because more thorough shuffles take more

time and reduce profits (and fortunately shuffling machines have not

yet caught on.)

I am assuming that the reader has some knowledge of blackjack and card

counting. A glossary of blackjack, card counting, and shuffle-tracking

terms can be found in the appendix.

THEORY AND EVIDENCE IN SUPPORT OF SHUFFLE-TRACKING

==================================================

Shuffle-tracking is based on a simple, sound theory, and there are

computer simulations and hand trials to back up the theory. Unfortunately,

shuffle-tracking is still in its infancy, so there is not a large body

of scientific literature on this subject. Therefore, my limited empirical

studies of shuffle-tracking may be the only such studies.

The Theory of Shuffle-Tracking

------------------------------

Shuffle-trackers actively exploit the inherent nonrandomness in casino

shuffles. The technique involves keeping track of the count in

different portions of the discarded cards, as they are observed during

play. In all the examples, you can consider the count to be high-low:

2,3,4,5,6 = +1 - 7,8,9 = 0 - 10,A = -1 - however, the tracking

explanations will hold for just about any counting system.

Almost all counts assign low/negative values to high cards (such as

10's are valued as -1) and high/positive values to low cards (such as

6's are valued as +1). It is a fact that high cards favor a player

and low cards favor the dealer; hence, removing a high card from the

shoe reduces the running count and removing a low card from the shoe

increases the running count. The higher the running count, the more

favorable (or less unfavorable) the game is for the player. True count

is running count divided by the number of unplayed decks, and for

the high-low system each unit of true count is worth .5% advantage.

As an example of a simple tracking method, if the end-of-shoe count is

-10, then you know that the count of the unplayed cards is +10. If the

unplayed cards all get shuffled into the top half of the shuffled

pile, where should you cut the cards?

The answer is right in the middle. The reason is that the unplayed cards

had a count of +10 - that means there were 10 more low cards than

high cards - you don't want to play those low cards which are now in

the top of the pile, so you cut in the middle to put at least some

of them out of play during the next shoe. You might also want to pat

yourself on the back and raise your betting during the first

half of the shoe, even though the count will probably start to go

negative. The reason is that on average the first half of the shoe

should now on average have a count of -10, which means there are

ten more high cards than low cards. An important thing to remember

during shuffle-tracking is that high count regions are bad and low

count regions are good. This can be counter-intuitive (no pun

intended).

There are two benefits to shuffle-tracking:

* knowledgeable cutting (removing low cards from play)

* additional information about regions that come into play

A Simple Shuffle-Tracking Strategy

----------------------------------

The previous section gave a trivial example of shuffle-tracking,

where you just use the running count at the end of the shoe. Tracking

more regions can give you more detailed information about the next

shoe.

Suppose it is a four deck game, with three decks actually dealt. You

record the running count for the first deck and call it A, the second

B, and the third C. (The recorded running count is for each deck

individually, so you must either take differences in the new and

previous running counts.) The unplayed deck is D, and it is assigned

the opposite of the final running count.

Suppose that shuffle starts by putting the unplayed cards on top of

the played cards. Then the pile looks like this:

D

C

B

A

And now if the pile is cut in two it looks like this:

B D

A C

And if the top halves are shuffled together and then the bottom halves

are shuffled together and placed on top, you wind up with this "profile":

A+C

B+D

The profile shows how the tracking units are combined. The plus sign

indicates that the estimated count in each two deck region is simply the

sum of two tracking units. So if A=-4, B=+2, C=+1, D=+1, then the

counts in each half of the shuffled shoe are estimated as follows:

-3

+3

Here, you would cut as close to the bottom, trying to keep the -3 in

front of the end-of-play card. You would bet more aggressively

in the -3 region and more conservatively elsewhere.

Empirical Results in Support of Shuffle-Tracking

------------------------------------------------

I ran my shuffle-tracking simulator on a casino shuffle, using

a realistic "clumpiness" of the riff and realistic inaccuracies in

pick sizes and randomness in plugging. Penetration was set at 66.7%

I ran the simulator on 100,000 shoes, and it was able to cut out an

average count of +7.0. Since 7 low cards were on average removed from the

first 5 1/3 decks of the shoe, this means that the true count at the

start of the shoe is effectively +7/5.3333 = +1.3, which is enough to

neutralize the base disadvantage of -.45% in the Atlantic City game

(or bad Nevada games). Shuffle-tracking also had an overall accuracy

in all regions of the shoe greater than the accuracy of true count

half way through the shoe. Thus, it could identify favorable

situations even at the start of the shoe, allowing the shuffle-tracker

to bet big off the top of the shoe intelligently, not just as

counter-camouflage. The complete results of these computer trials are

given in an appendix.

In hand-done trials (which used a different casino shuffle),

shuffle-tracking had a higher % advantage than card counting alone,

statistically significant to the 90% confidence level. Also, while

shuffle-tracking, I cut out more low cards than high cards,

statistically significant to the 99.5% confidence level. The complete

results of the hand-done trials are given in an appendix.

I also ran a full-blown computer simulation of shuffle-tracking

an Atlantic City shuffle (the Random Pick Order Six Zone Shuffle)

with AC rules plus late surrender, 75% penetration. A great

deal of randomness was put into the shuffle, making it difficult

to track, and the tracking and card counting was not done perfectly.

The shuffle-tracker was given the cut card every time, however.

With a 1-8 spread, never abandoning the table, it achieved a 1.0%

advantage. In constrast, a simulated regular card counter did

little better than break even in this game, if not permitted to

abandon negative counts. Thus it would appear that shuffle-tracking

provided a gain of nearly 1% here, but this is a tentative conclusion.

By abandoning hopeless shoes, the shuffle-tracker's advantage

could be increased - a regular card counter gains about 0.5%

by abandoning true counts of -1 or worse on the AC game, and

a shuffle-tracker has a much better of when a shoe is hopeless

than a regular card counter.

THE PRAGMATICS OF SHUFFLE-TRACKING

==================================

All this is well and good, but how can it work in practice? After all,

a casinos won't allow you whip out paper and pencil (or your

shuffle-tracking computer) at the blackjack tables! Shuffle-tracking

requires a lot of "table smarts", just like card counting. You not

only have to know how to shuffle-track well, but you also have to

know how to avoid detection by casino personnel.

How to Track Without Going Crazy

--------------------------------

In place of paper and pencil or computers, shuffle-trackers use their

chips to provide "memory". You can use clock notation to represent

0 (12 o'clock) through 11 (11 o'clock). You can use different

colors to represent positive versus negative or 0-11 versus 10-21. Or

you can use a single color with clock notation running from 0 to +5 to

clockwise and 0 to -5 counterclockwise (6 o'clock is then not used.)

You can use patterns in how the chips are stacked, perhaps offset

to the left or right. Whatever. Obviously, you must be discrete, but

many gamblers play with their chips.

A shuffle-tracker who is playing through the second shoe at a table

will have four groups of chips: betting chips, count record chips,

running count chips, and shuffle-track prediction chips. The betting

chips are an unorganized mess from which all bets are placed and into

which all winnings are placed. The count record chips are the counts

of various regions in the current shoe. The running count chips denote

the running count when the last count record chip was placed. And the

shuffle-track prediction chips are the predictions of the counts in

the current shoe.

The chip notation is used to record the numbers, such as counts in

various regions of the shoe. For example, suppose that a four deck

shoe is being tracked with four regions, A, B, C, and D (the latter

being the unplayed cards). After the first deck (A), you place a chip

to denote the running count. After the second deck (B), you subtract

the current running count from the previous running count. You stack

this chip on top of the previous and then record the current running

count separately. For the next deck (C) you take the count difference

and stack a chip representing this on the count record pile while also

updating the record of the current running count. At the end of the

played portion of the shoe, you take the opposite of the running count

and assign this to D. (If there were more than one tracking unit that

was unplayed, then this final count would be split among the unplayed

tracking units as an estimate. If you are really sharp, you can split

this count unevenly according to previous tracking information.)

Shuffle!

For casinos that use the same exact shuffle each time (with no randomness

in the order of picks or plugging), you can analyze the

shuffle away from the tables and come up with a "profile". This is
just a

precomputed diagram showing how to combine different portions of

the shoe. A profile was listed in a previous section that look like this:

A+C

B+D

This profile can be memorized and a small cheat sheet of it (perhaps on

the back of a business card) brought to the casino in case you freeze

under pressure.

For shuffles where the dealer has some randomization effect, like

mixing up the order of picks, the tracking requires more of a brute

force approach. Using brute force is simpler, but also less

disguised. Here, you actually "shuffle" your chips the same way in

which the dealer shuffles the cards. An example of this is given

in a later section on the Random Pick Order Six Zone Shuffle.

As the shoe is played through, the shuffle-track prediction pile(s) shrink

and the count record chip pile(s) grow. One should make a mental

note of how close the newly recorded counts are to the estimates, and

also to compare this to the prediction of the true count (i.e.,

the *opposite* of the true count.) Although you can't expect

shuffle-tracking to be anywhere near 100% accurate in terms of

sign much less magnitude of the count, you should be able to observe

a correlation. If the shuffle-track predictions do not seem correlated

to the observed counts, then you may be making mistakes or the shuffle

may not be very trackable.

Fine Points of Shuffle-Tracking

-------------------------------

Always remember that shuffle-tracking is not mutually exclusive to

card counting. You can still bet according to true count. However,

tracking gives you additional information that will either allow you

to raise your bets more often or more safely or perhaps both. One

possibility is to go to a higher bet when either the shuffle-track

OR the true count indicate that this is a good idea. The other possibility

is to go to a higher bet only when both the shuffle-track AND the

true count agree that this is a good idea. Actually, you should listen to

shuffle-track predictions more towards the beginning of the shoe, and

true count more towards the end of the shoe, because true count is

of no help at the beginning of the shoe but is very accurate at the

end of the shoe.

Deciding what tracking units to use is important. Generally, the

tracking units relate to the dealer's pick sizes, otherwise the

tracking predictions may be unnecessarily inaccurate. Also, if you choose

too small tracking units, you will not be able to "eyeball" the discard

tray to determine which tracking unit you're in, but if you choose too

large tracking units, you may have insufficient information to give

you much of an edge.

Shuffle-tracking teams can be effective. For shuffles that can

be profiled, each team member can be responsible for generating a

tracking prediction of some portion of the shoe, thus splitting the

mental burden of shuffle-tracking. One team member can be responsible

for just counting the number of cards that have been put into the

discard tray and signaling the other members as the last few cards of each

tracking unit get discarded. Teams can adjust the number of hands they

play in order to make end of tracking units coincide with the end of a

round (when lots of cards go into the discard tray at once.) Four or

more team members can completely take over a seven spot table (each

playing one or two hands), giving the team complete control over the

cut card. This allows the team to build up a large clump of low cards

that can be consistently cut out of play. However, four skilled

shuffle-trackers might very well be better off each playing solo

off of a pooled bankroll, so don't play at the same table unless

you think there is enough of a benefit to outweigh the increased

variance (the hands are correlated with each other at the same

table, since all depend on the same dealer's hand.)

Intelligent cutting is one of the benefits of shuffle-tracking.

Sitting at third base gives a shuffle-tracker an advantage, because

it increases the likelihood of getting the cut card in casinos where

the cut card is given to the third base player if it comes out while

the dealer is resolving his own hand. (It's also nice to sit at third

base, because there's usually lots of room there to spread one's chips

out for shuffle-tracking purposes, plus third base has a slightly

higher advantage for regular card counters anyway, due to more cards

being seen before the player makes his plays.) A shuffle-tracker

can also spread to two hands at the end of a shoe to boost the odds of

getting dealt the cut card. Using both these techniques at a full

7-spot table would give one over a 3/7 chance of getting the cut card.

One can also often obtain the cut card simply by asking for it.

Saying something like "I feel lucky - how about letting me cut for

us?" usually does the trick. Players are usually either antipathetic

or nervous about cutting the cards, so they will generally relinquish

the cut card gladly. One should use some restraint in doing

this while a pit critter is lurking nearby, however. You can attempt

to tell people where to cut, but this is harder than it sounds, so

it's better just to get the cut card yourself.

Often players leave during the shuffle, so be on the look-out for

an abandoned cut card. If the player who had the cut card leaves,

then pounce on it or ask another player to pass it to you.

Sometimes the shuffle-track predictions will not reveal any good place

to cut the cards. At such times, obviously you don't need to fight to

get the cut card.

Even if someone else cuts, you can still judge how good their cut is

and decide whether or not to remain at the table. If there is an

obvious good region that will be in play, it may be worth staying

to bet big in that region even if another good region was cut out,

yielding an overall bad shoe. However, always remember that if you

leave to go to another table, on average an equal number of good

cards and bad cards will have been cut out, whereas if you stay with

a bad cut, then you are pretty sure that more good cards have been cut

out than bad cards.

How to Avoid Casino Countermeasures

-----------------------------------

The harshest casino countermeasure is that of barring. It is not

illegal to count cards or shuffle-track. However, it is illegal in

Nevada to enter a casino once you have been barred; if you do, you may

go directly to jail. In New Jersey, you can be consoled that the New

Jersey Supreme Court outlawed the practice of barring skilled

blackjack players. Thus, Atlantic City is somewhat less aggressive

about intimidating counters, especially because they've made the

Atlantic City game so poor that it's not really worth a card counter's

time!

This is not to say that Atlantic City casinos do not care about card

counters. They are very paranoid about them (which is not justified

given the poor games) and they can and do take other countermeasures.

Also, Nevada casinos will activate many other countermeasures before

resorting to barring. The simplest and most effective countermeasure

is the "shuffle up", when the cards are prematurely shuffled. If you

place a large bet, they may shuffle up, or if the dealer is card

counting too, he may shuffle away *any* favorable situation!

This is very rare in Atlantic City, where the preferred countermeasure

is to move the end-of-play card to perhaps the 50% point after the

next shuffle. In Nevada, you will usually be kindly asked to play

craps or any other game than blackjack before you are actually barred.

The casinos do not part easily with their money. In fact, I have seen

many pit bosses get upset when someone (whom I can tell is about as

intelligent as a squid) happens to get lucky and walks away from the

table with a lot of money. The variance is so high in blackjack that

that a very good counter can easily take a big loss for eight hours,

while the squid keeps raking in the bucks, but this is something that

card counters and casino employees do not in general appreciate fully.

In the long run, of course, the counter grinds out a profit, while

the squid will eventually lose everything unless it slinks away from

the table first.

Since the average pit critter does not understand these facts of

statistics, you must appear to lose and take little if any chips away

from the table. This is difficult, since in order to shuffle-track

you may need dozens of chips on the table at all times. Even if you

buy in for the same amount that you leave with, pit bosses may get

upset if you walk away from the table with a few hundred dollars in

chips.

One method of hiding a win is to simply not let the dealer color you up.

"Coloring up" or "coloring in" is where the dealer counts
your chips

and gives you a few higher denomination chips. If you don't let them

color up your chips, then they will be uncertain as to how much you

take away from the table. However, this itself will raise suspicions,

especially in Atlantic City, where the casinos are quite insistent

about coloring up a player's chips upon his leaving a table. Another

method to disguise your winnings is to pocket chips *discretely*. The

best way is to pick up a stack of chips and hold it in your hands to

bet with for a few rounds - then when no one is looking and your hands

are clasped around the chips, drop your hands off the table and shove

the chips into your pocket while appearing to be interested in a

cocktail waiter/waitress or something. Only do this if there are

other players taking that denomination of chips away from the table,

because the pit critters keep careful records of the chip trays.

Real gamblers often play with their chips. You should practice at home

with real casino chips to learn to figit with your chips most of the

time and to disguise your occasional moves to record information with

some of your chips. It helps to be messy and careless with your

chips. Leaning over your chips with your arms above them will help

obscure the information in your chips from the eye in the sky and the

pit boss' regular rounds. (Side note: gosh how I hate "leaners" when

I'm trying to back-count - it makes it hard to see all the cards from

behind!)

Remember that there is little risk of them mistaking you for a normal

card counter, because you will sometimes be able to bet high off the

shoe and high in the middle of the shoe in opposition to the true

count. Thus, shuffle-tracking itself is a means of disguise, so long

as you don't appear to be a little too interested in your chips.

REAL WORLD SHUFFLE-TRACKING STRATEGIES

======================================

The simple shuffle-tracking strategy described previously was just an

example. Casinos do not usually use so simple a shuffle. Different

shuffles will require different shuffle-tracking strategies.

Many of the casinos use shuffles devised to frustrate

shuffle-trackers.

There appear to be two shuffle techniques designed to frustrate

shuffle-trackers. One is to introduce some dealer-driven randomness.

This includes plugging unplayed cards into played cards at several

random locations, and random orders of picks for zone shuffles.

The other tracking countermeasure is the stutter shuffle, where

two piles are shuffled by taking a pick from one pile and a pick

from the already shuffled cards and putting the result on top of the

already shuffled cards (the stutter pile), and then taking a pick from the

other pile and a pick from the already shuffled cards and putting the

shuffled result on top of the already shuffled cards, and so on.

It is interesting to note that I have never seen a casino shuffle that

uses *both* of dealer-driven randomness and the stutter shuffle; it seems

that the casinos feel that either one alone is sufficient to thwart

shuffle-trackers. In truth, they are pretty much correct about the

stutter shuffle,but sometimes the dealer dealer-driven randomness

techniques can be conquered.

Each casino generally alters its shuffle every few months. Again, this

is obviously intended to thwart long-term attacks from shuffle-trackers.

There are two main approaches one can take to analyze a shuffle. The

most straightforward is to take eight decks of cards (or however many

the casino uses), label the backs with letters denoting different

tracking units, sort them according to tracking units, and then

shuffle them as the casino does. You then count the numbers of cards

from each tracking unit that wound up in each region.

The other way analyze a shuffle is to take a more symbolic

approach. Start out with an equal number of copies of each tracking

unit letter. (With the number of "copies" just being a convenient

number that will avoid fractions of tracking units during the analysis.)

For example, if you have four tracking units and two copies of each,

then the cards might look like this before shuffling:

D

D

C

C

B

B

A

A

You then manipulate this symbolically to arrive at the distribution of

cards after the shuffle. This is the technique used previously in

generating a "profile" of a simple shuffle, and this techique will

also be used in the following sections on specific casino shuffles.

The information here may be slightly dated - my last full survey

of AC shuffles was around January 1991.

The Criss-Cross Zone Shuffle

----------------------------

The Claridge, Trump Castle, and the Sands in Atlantic City each perform

what I call the Criss-Cross Zone Shuffle. There are slight differences

that will be explained.

First, the Claridge plugs the unplayed cards into the discards (at

three or four locations), while the Sands merely places the unplayed

cards on top of the discards.

For the Sands/Castle version, take the top ~2 decks off the top of the

discard pile (i.e., pretty much just the unplayed cards) to form what

pile #3, and *then* cut the rest of the discard pile in half to the

right to form piles #1 and #2. For the Claridge, on the other hand,

cut initial pile in half to dealer's right, and call the piles #1

(bottom cards) and #2 (top cards). Pick .5 [variant: up to 1] decks

off each of #1 and #2, and join the picks to form #3.

Cut #1 and #2 each in half, dropping the picked piles away [variant:

toward] the dealer. Call the new piles #1a (bottom of #1), #1b (top of

#1), #2a (bottom of #2), and #2b (top of #2). From now on, picks from #1a,

#1b, #2a, and #2b will all be 1/3 [variant: 1/4, sometimes even less] of

these piles, and picks from #3 will be 1/6 [variant: 1/8, sometimes

even less] of this pile. Join picks from #2a, #1b, #3 (with #3 always

on top) - riff, riff, put on done pile. Join #2b, #1a, #3 - riff, riff,

put on done pile. Now back to #2a, #1b, #3, and continue alternating

until all cards are in done pile. This results in 6 [variant: 8,

sometimes even more] shuffled regions in the done pile.

This shuffle is the same for the four and six deckers as the eight

deckers, except that picks are scaled down appropriately.

If you can find a dealer that is fairly consistent, then you can

devise a profile of how this shuffle combines tracking units in

the next shoe. Unfortunately, having that extra C pile leads to either

a more complicated or less accurate (or both) profile.

Here's how to make such a profile using a symbolic approach. First,

we start with, say, three copies of each of six tracking units (A-F).

Then the initial cards look like this, assuming 1/3 of the cards

(tracking units E and F) are undealt:

D

D

D

C

C

C

B E

B E

B E

A F

A F

A F

For the Sands/Castle shuffle, the undealt cards are placed on top of the

played cards and then taken right back off to form pile C. Then

The played cards are broken into four piles:

(1b) (2b)

B D

B D (3)

B D E

E

(1a) (2a) E

A C F

A C F

A C F

The first part of the shuffle takes picks from #1b, #2a, and #3, the

second takes picks from #1a, #2b, and #3, and so on to yield:

A+D+F

B+C+F

A+D+F

B+C+E

A+D+E

B+C+E

The tracking predictions should technically be divided by three. Also,

tracking units E and F are the unplayed cards, and so they can be

estimated to have the same count. Thus, a possible simplification might

be to just ignore E and F since these counts are distributed evenly through

the shoe.

The Random Pick Order Six Zone Shuffle

--------------------------------------

I will now describe my shuffle-tracking strategy for the eight deck

shuffles used by Resorts and TropWorld at the time that this was

written. This same shuffle is also used on the 8 deck shoes with

over/under at Tropicana in Las Vegas. First a description of the

shuffle, which I call the Random Pick Order Six Zone Shuffle...

The dealer plugs the unplayed cards into discards in three spots,

usually one deck up from the bottom, the middle, and one deck down from

the top. The pile is broken in two (to the right usually). Then each

pile is broken in three, to create a line of six roughly equal piles.

If the original pile after plugging looked like this:

F

E

D

C

B

A

Then the resulting six piles would look like from the dealer's perspective:

C B A D E F

Except that female dealers (with small hands) sometimes do this:

B C A D F E

A 1/2 pile pick is made randomly from A, B, or C and riffed

with a random pick from D, E, or F. The just riffed cards are then cut

in two and riffed again. This is repeated until all the cards have been

shuffled. Usually, the dealer does not take the second pick from a pile

until all other piles have had their first pick, but this is not always true.

The result of this shuffle is quite nonrandom, and my shuffle-tracking

simulator showed that it would be quite trackable, if only you can

combine the appropriate tracking units. For example, suppose that the

dealer always picked in the order A+D, C+F, B+E, A+D, C+F, B+E. Then

the result would be this:

B+E

C+F

A+D

B+E

C+F

A+D

Easy, huh? Or if we break down the observed cards more finely, using A1

to stand for the bottom part of A and A2 for the top of A, and analogously

for the rest, we start with this:

F2

F1

E2

E1

D2

D1

C2

C1

B2

B1

A2

A1

And after being shuffled this becomes...

B1+E1

C1+F1

A1+D1

B2+E2

C2+F2

A2+D2

So, if the count for B1 were -5 and the count for E1 were -2, then we

can estimate that the first sixth of the shoe shuffled as above would

have a count of -7.

Now the above letters sort of assumed no plugging. A plugged pile

might start like this, if the unplayed cards are units E2, F1, and F2:

E1

D2

F2 <- PLUG

D1

C2

C1

F1 <- PLUG

B2

B1

E2 <- PLUG

A2

A1

And the above would then wind up like this:

E1+F1

D2+B2

F2+B1

D1+E2

C2+A2

C1+A1

In practice, the individual counts of the plugged cards (tracking units

E2, F1, and F2) are not generally known, so the end-of-shoe count can

be split between them as an estimate (possibly biased by previous

tracking information.) Also, the above profile cannot be used in

general, since the order of the picks and placement of the plugs

varies.

But that's okay. We can use brute force.

If penetration is 75%, then you've got a stack of nine chips

representing the played cards and a stack of three chips representing

the unplayed cards. This is convenient for the above shuffle, because

the unplayed cards are split into three and plugged into the played

cards. A brute force approach is required to track this. Watch the dealer

carefully, and just plug your chips wherever he plugs his cards! Cut

the chips in half the same direction (mirrored, so usually you cut to

the left) that the dealer cuts the cards. Break the chips into six piles

of two chips in the same pattern as the dealer breaks the cards six

piles. You can be a little less conspicuous by not doing these

activities exactly when the dealer does them. For example, I usually

cut my chips 5-4 before the dealer and I do the plugging, and I break

mine into the smaller piles before he does. I just have to keep one

eye on the dealer to make sure he is doing the normal routine. Then,

as the dealer takes a pick from each of two piles, I take a chip from

each of my corresponding piles (mirrored). I put these into two

separate piles. The next picks are stacked on top of these piles.

The end result is two piles of six chips. At each level, the sum of

the two chips is the estimate of the count in that region. As I put

down the chips, I note where excessively positive (bad) and negative

(good) regions are. If I get the cut card, I then attempt to cut

just below the worst regions in order to cut them out of play. If

someone else is cutting, I note where they cut. I then discretely

(no hurry) cut my chips and restack them. The top chips on the two

piles added together are then an estimate of the first 1/6'th of the

shoe. I can adjust my betting and playing appropriately. I don't

consider the information to be very reliable unless both chips are

pointing strongly in the same direction - then chances are very good

the shuffle-tracking has at least the sign of the count in that region

correct.

The (Dreaded) Stutter Shuffle

-----------------------------

Bally's Park Place and Bally's Grand use the Stutter Shuffle. This

shuffle is to shuffle-trackers what Kryptonite is to Superman. It

is not totally random, but it is not worth tracking.

Place unplayed cards on top of played cards. Split the eight decks

into two ~four deck piles, call them #1 (bottom) and #2 (top). Picks

are taken from each pile, shuffled once, and placed in the "stutter

pile", #3. Then a pick is taken from one of the piles (usually #1) and

shuffled with a pick from #3. The result is placed *under* #3. A pick is

taken from the other pile and shuffled with a pick from #3, and the

result is again placed *under* #3. After this point, it continues to

alternate between #1 and #2 (both with picks from #3), but the results

are placed on *top* of #3; the shuffle proceeds until all the cards are

in #3.

The dealers at Bally's Grand tend to use very large pick sizes, but

in the following profile, I will assume that the pick sizes are just

1/2 deck, so that this will be compatible with the following section

on the Stutter Plus Shuffle. Here is how the profile looks for

this stutter with sixteen regions from A (bottom) to P (top), where

(x y) is defined as the average of the counts of regions x and y:

1. (I (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))))

2. (I (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))))

3. (A (J (B (K (C (L (D (M (E (N (F (G (H P)))))))))))))

4. (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))

5. (B (K (C (L (D (M (E (N (F (G (H P)))))))))))

6. (K (C (L (D (M (E (N (F (G (H P))))))))))

7. (C (L (D (M (E (N (F (G (H P)))))))))

8. (L (D (M (E (N (F (G (H P))))))))

9. (D (M (E (N (F (G (H P)))))))

10. (M (E (N (F (G (H P))))))

11. (E (N (F (G (H P)))))

12. (N (F (G (H P))))

13. (F (G (H P)))

14. (G (H P))

15. (O (H P))

16. (O (H P))

For example, the bottom deck (regions #15 and #16) is composed of

1/2 region O and 1/4 each of regions H and P. Obviously, this

is not a practical prediction scheme, especially with respect

to the top deck. This could be simplified by using larger tracking

units; for example, there could be just four tracking units: A/B/C/D,

E/F/G/H, I/J/K/L, and M/N/O/P. But even this would be complex. Other

simplifications could be made, but only at the cost of much accuracy. So

long as there exist zone shuffles, the shuffle-trackers should avoid

stutter shuffles like the plague!

The Stutter Plus Shuffle

------------------------

As if the above Stutter Shuffle weren't enough, several Atlantic

City casinos go further, namely Taj Mahal, Showboat, and Trump Plaza.

In doing so, they are complying with the regulations that require a

reasonably random shuffle, but I doubt if their motive is to obey the

laws. The cards are broken into two four deck piles again (A and B),

and half deck picks are taken from each, shuffled, and then cut in

half, with each half deck being placed in a separate "done pile" (C

and D). This continues until all the cards are in the two done piles.

The done piles (C and D) are then stacked on top of each other, and

that's it. The notation for the profile below is the same as in the

previous section, but note that the entries take up two lines, since

they are so long:

1. ((L (D (M (E (N (F (G (H P))))))))

(O (H P)))

2. ((C (L (D (M (E (N (F (G (H P)))))))))

(O (H P)))

3.((K (C (L (D (M (E (N (F (G (H P))))))))))

(G (H P)))

4. ((B (K (C (L (D (M (E (N (F (G (H P)))))))))))

(F (G (H P))))

5. ((J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))

(N (F (G (H P)))))

6. ((A (J (B (K (C (L (D (M (E (N (F (G (H P)))))))))))))

(E (N (F (G (H P))))))

7. ((I (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))))

(M (E (N (F (G (H P)))))))

8. ((I (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))))

(D (M (E (N (F (G (H P)))))))

9. ((L (D (M (E (N (F (G (H P))))))))

(O (H P)))

10. ((C (L (D (M (E (N (F (G (H P)))))))))

(O (H P)))

11. ((K (C (L (D (M (E (N (F (G (H P))))))))))

(G (H P)))

12. ((B (K (C (L (D (M (E (N (F (G (H P)))))))))))

(F (G (H P))))

13. ((J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))

(N (F (G (H P)))))

14. ((A (J (B (K (C (L (D (M (E (N (F (G (H P)))))))))))))

(E (N (F (G (H P))))))

15. ((I (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))))

(M (E (N (F (G (H P)))))))

16. ((I (A (J (B (K (C (L (D (M (E (N (F (G (H P))))))))))))))

(D (M (E (N (F (G (H P)))))))

Note that the pattern for 1-8 is the same as the one for 9-16.

What the above mess means is that most cards have a chance of being

*almost* anywhere in the resulting shoe. It is effectively not

trackable, especially considering that the randomness of the

dealer's pick size and riff will considerably alter the distribution

of cards. Avoid such thorough shuffles if at all possible.

OTHER SOURCES OF SHUFFLE-TRACKING INFORMATION

=============================================

There is an article that appeared in the New York Times in which a

mathematician had proved that it takes 7 imperfect "riff" shuffles

to randomly order a single deck, and many more to randomly order

multiple decks. The casinos can't afford to shuffle this much. However,

nonrandomness isn't necessarily a bad thing. In fact, Snyder in Blackjack

Forum has shown empirically with computer simulations that very nonrandom

shuffles can help basic strategy players by a few tenths of a percent

(yielding a positive expectation game in extreme cases) and reduce

the profits of card counters by only a tenth of a percent at most

(and probably no where near that much.)

In "Break the Dealer", Patterson and Olsen published the first book

describing shuffle-tracking. It is still the only book whose primary

focus is on shuffle-tracking. This is unfortunate, since the book

serves more as an advertising tease for their TARGET system than as

a treatise on shuffle-tracking. (TARGET is a non-counting blackjack

system that has been criticized by a number of blackjack experts as

having no scientific basis or empirical proof of its effectiveness.)

Patterson and Olsen describe shuffle-tracking in nice graphical terms,

but don't go into much detail. In general they believe that certain

shuffles bias the cards for or against the player, and there just

isn't any evidence to support this claim (and quite a bit to refute it.)

In particular, they fear the strip shuffle (where the order of the cards

is reversed with a rapid series of pick), and there is no reason to

fear this shuffle, unless you are trying to shuffle track and the strip

mixes up your tracking units. Also, they fear like-card clumping (the

natural tendency of similar cards to cluster given the order they are

discarded), and this is not something to be worried about. In sum,

be very skeptical about anything these authors say, though they are

not *always* wrong.

Mason Malmuth has the only other publication speaking of

shuffle-tracking in any depth, "Blackjack Essays". (He calls

shuffle-tracking "card domination".) He is very enthusiastic about
it

- perhaps overly so, since he estimates an expected win rate of 4.5%,

which is very unlikely. Mason added a note in which he backs down from

this estimate on the basis that the *actual* advantage in blackjack is

rarely 4.5% and shuffle-tracking will not even identify all of these

situations. Mason recommends that the shuffle-tracker not try to keep

very detailed information about the deck composition; this is not my

philosophy, and perhaps Mason is wrong about this too.

Shuffle-tracking is explained briefly in Zender's "Card Counting for

the Casino Executive", and it is noted that dealers can employ

shuffle-tracking in reverse, to shuffle the good cards to where

they will be cut out of play.

Blackjack magazines occasionally have information on shuffle-tracking.

Snyder's "Blackjack Forum" and Olsen's "Blackjack Confidential"
have

both mentioned shuffle-tracking in certain articles, though I have not

seen any in-depth analyses of shuffle-tracking in these magazines.

Appendices of the Blackjack Shuffle-Tracking Treatise

Copyright 1990, Michael R. Hall

APPENDIX I

Glossary of Terms

General

-------

shoe - that thing used to hold multiple decks.

end-of-play card - that colored card inserted into the shoe; when it is

dealt, that is the last round before the shuffle. It

is inserted by the dealer after the shuffle, usually

at most 80% down from the top, to thwart card-counters.

cut card - this is physically the same as the end-of-play card, but when

a player is dealet the end-of-play card, then they get to cut

the cards. The player inserts the edge of the card, and the

dealer physically cuts the cards and restacks them and begins

play.

burn card - the card that is burned (discarded) at the beginning of the shoe,

probably to thwart people who try to cut themselves a bent card.

played cards - the cards that are played and discarded.

unplayed cards - the cards that are still in the shoe.

like-card-clumping - the clumping of low cards with low cards and high

cards with high cards that occurs naturally as a result

of the order in which cards are discarded as well

the fact that you generally stand if you have two

high cards but hit if you have a bunch of low cards.

bankroll - cash on hand for gambling.

Card-Counting

-------------

card-counting - the strategy of keeping the running count, among other things.

card counter - one who performs card-counting.

running count - a number (usually integer) representing how many more high

cards than low cards have been observed.

count - see running count.

true count - running count divided by number of remaining decks.

Shuffle-Tracking

----------------

shuffle-tracking - the strategy of noting the count in tracking units

throughout the played portion of the shoe and then

averaging the tracking units that are shuffled together

to form tracking predictions of regions.

shuffle-tracker - a card counter who performs shuffle-tracking.

tracking unit - a number of cards that are tracked (i.e., counted) as one unit.

tracking prediction - prediction of the count of a region accomplished

via tracking (my term).

region - a number of cards whose count is predicted by tracking;

usually a multiple of tracking units (my term).

Shuffles

--------

riff - standard shuffle where you take two piles and bend the corners up,

letting them fall together, ideally alternating perfectly.

interlace - see riff.

strip - reversing the order of the cards - can be done with individual

cards or clump-by-clump. This is never done alone, but often

accompanies a riff.

zone - a quick method of shuffling multiple decks by separating the cards

into several zones and shuffling these zones together.

stutter - a slow method of shuffling multiple decks that insures that any

card has a chance of being anywhere in the result; each shuffling

consists both of unshuffled cards and already shuffled cards.

This shuffle is probably done solely to thwart shuffle-trackers.

plugging - a method of putting unplayed cards into several locations

in the middle of the played cards. This is probably done soley

to thwart shuffle-trackers.

APPENDIX II

Hand Trial Shuffle-Tracking Empirical Results

Experimental Procedure

----------------------

I played eight deck blackjack with AC rules, late surrender, and the

Claridge shuffle to simulate conditions at the Claridge. This was all

with real cards, not computer bits. There were four simulated extra players,

and I took the "middle seat" and played either one or two hands, depending

on the count. I shuffle-tracked by noting the High-Low count in various

regions of the shoe, and then used a shuffle tracking profile similar to

the one given in the section on the Cross-Cross Zone Shuffle. I used this

to decide where to cut the cards, trying to cut out the positive count

regions (which have more low cards than high cards). I placed the

end-of-play card two decks from the end.

Trials

------

+8 +9 +5 -2 +8 0 +3 +3 +7 +8 -11 +12 +19 +3 +9 -5 -10 +11 -5 +3 +3 -8

-5 -10 +1 +8 +9 +2 +17

+6 +9 +5 +7

Each of the above numbers is the number of extra low cards versus high

cards left *unplayed*. (This is not quite the same as the cards behind

the cut card, since the cards after the cut card are used to finish

the round in which the cut card appears.) It is the additive inverse of

the count at the end of the shoe (though I counted the values of the

remaining cards at the end to reduce the chance of errors in the statistics.)

The trials are broken into three lines to note when I switched shuffle

tracking strategies. The second line was intended to be an improvement

over the first, and the third was intended to be an improvement over the

second - each improvement involved switching to a finer grain of

tracking units.

Analysis

--------

Okay, we're gonna do a test of hypotheses on the mean of a normally

distributed variable whose variance is unknown. The null hypothesis

is that cutting "intelligently" by shuffle tracking should leave on

average a count of 0 unplayed, just as one would expect with a random

cut (concerns about position of cut card and low cards being used

up more quickly notwithstanding.) The alternative hypothesis is

that cutting by shuffle-tracking should leave a positive count

of cards unplayed. Oh yeah, we can assume that the distribution

is normal, because at least in the null hypothesis case, the data should

follow a beautiful normal distribution, centered at 0 (think about it).

Let u0 = 0 be the mean of the null hypothesis distribution

Let u be the mean of the alternative hypothesis distribution

Let n be the number of trials (shoes)

Let x be the average count of the unplayed cards

let s^2 be the variance in the count of unplayed cards

H0: u = u0

H1: u > u0

s^2 = 53.45

n = 33

x = 3.606

We should reject H0 if in LISPish notation...

(/ (/ (- x u0) (/ s (sqrt n)))) > t(alpha, n-1)

Plugging in we get,

(/ (- 3.606 0) (/ (sqrt 53.45) (sqrt 33))) = 2.8334

Looking in a table, we find,

t(.005, 30) = 2.750 and t(.0025,30) = 3.030

Thus, there is significant difference, up to the 99.5% confidence level,

assuming I did all the computations correctly. Therefore, we should reject

the null hypothesis that shuffle tracking does not work, and hence we must

believe that it does work, at least for me in my own home. By "work",
I

mean that you can certainly cut out, on average, more low cards than high

cards. This effectively raises your running count!

Now, if we assume that the above average of 3.6 more low cards than high

cards is a realistic average (warning: the 99.5% confidence does not apply

to this assumption), then that implies if I shuffle track at the Claridge

and cut the cards myself, then the running count is *effectively* 3.6 points

higher than the actual running count. The commonly quoted number in blackjack

books is that one true count point of High-Low is worth .5% advantage. Thus,

in the played portion of the shoe, my gain from shuffle tracking is

(* (/ 3.6 6) .5%) = 0.3%. Note that this benefits everyone at the table;

stupid players and basic strategy players will get this percentage increase;

card counters will win an extra amount with reduced risk; and I will argue

that shuffle tracking counters will win much more at even lower risk,

because of the benefits of having information about the distribution of

high and low cards in the shoe.

I argue that my gains from shuffle tracking could potentially be much

higher than 0.225%-0.9%, because not only do I have the benefit of

cutting out the low cards, but also shuffle tracking provides a rough

profile of the clumps of high cards (and low cards) in the played part

of the shoe; this indicator coupled with the true count is very powerful.

(How many times have you raised your bet on a high count, only to have

the count go still higher while you lose? This doesn't happen often with

shuffle tracking.) This local information I believe is worth more

than cutting out the low cards, so even when I don't get the cut card

or when I accidentally cut out more high cards than low cards, I can

still benefit greatly from shuffle tracking.

Shuffle tracking doesn't always work. The inherent randomness sometimes

makes the technique backfire. But it works on average to cut out the

low cards as the above statistics have shown.

Bankroll Data

-------------

For whatever it is worth, here is a record of my bankroll for the

home trials. The first series is with no shuffle tracking. As you

can see, it starts at $300.00 and ends up at $297.50 - rather depressing,

but not abnormal, since the AC game is depressingly close to even for card

counters with only a 1-~4 betting spread. The second series is with

shuffle tracking and also starts at $300.00, but ends up at $720.00

See my previous post for the explanations of playing conditions. Minimum

bets are always $5, and the maximums used within a particular shoe are

listed below by the bankroll. When the max bets are on two hands, the bets are

listed with a plus sign in between. Note that sometimes the bets on two

hands are not equal, since I was trying for a bet size in between (and

besides it's good cover in an actual casino.) The recorded bankrolls are

*after* a given shoe has been played. I estimate that 25 rounds were

played per shoe, and I averaged maybe 1.25 hands at a time and a total

bet average of maybe $10. By the way, I believe the relatively low

observed variance is due largely to late surrender (this is a benefit

that few people mention about this rule.)

WITHOUT SHUFFLE TRACK WITH SHUFFLE TRACK

###################### ##########################

SHOE BANKROLL MAX BET BANKROLL MAX BET CUT-OUT

---- -------- ------- -------- ------- -------

- 300.00 - 300.00 - -

1 300.00 5 347.50 5+5 +8 STARTED TRACKING METHOD I

2 340.00 5+5 315.00 5 +9

3 320.00 5 320.00 5+5 +5

4 347.50 5 300.00 5 -2

5 377.50 5 375.00 10+10 +8

6 382.50 10+10 342.50 5 0

7 390.00 10+10 395.00 5+5 +3

8 387.50 10+10 407.50 5 +3

9 372.50 5 415.00 5 +7

10 402.50 5 427.50 5 +8

11 402.50 5+5 425.00 5 -11

12 407.50 5 422.50 15+15 +12

13 412.50 5 432.50 5+10 +19

14 410.00 10+10 395.00 5 +3

15 412.50 5+5 345.00 5+5 +9

16 367.50 5+5 327.50 5 -5

17 342.50 5 322.50 5+10 -10

18 327.50 15+15 412.50 10+10 +11

19 335.00 5+5 405.00 5+5 -5

20 322.50 5 375.00 5 +3

21 315.00 5+10 375.00 5+5 +3

22 337.50 5 382.50 5+5 -8

23 330.00 5 465.00 10+10 -5 STARTED TRACKING METHOD II

24 335.00 5+5 565.00 15+15 -10

25 327.50 5 615.00 5+5 +1

26 337.50 5 595.00 5 +8

27 282.50 10+10 597.50 5 +9

28 290.00 5+5 607.50 5 +2

29 297.50 5 590.00 5 +17

30 570.00 5 +6 STARTED TRACKING METHOD III

31 572.50 5+5 +9

32 732.50 10+10 +5

33 720.00 5+5 +7

As you can see there is an impressive bottom line difference ($2.50 loss

versus $420.00 gain) between the "without tracking" and then "with
tracking"

play records. Note that the earnings in the shuffle tracking column don't

really take off until the shoes where tracking methods II and III are used.

This is probably because "method I" did not provide a detailed profile
for

the shoe - it only indicated where to cut - whereas the other methods gave

me a reasonable idea where to find the clumps of high versus low cards.

Unfortunately, there were relatively few trials involvings these

advanced tracking schemes.

Is the difference statistically significant?

I played about thousand hands for each series (about 906 for without

"column", and 1031 for the "with" column). 1000 trials is
"starting" to

be a signicant number, though often millions or billions of trials are

required to get above the inherent "noise" in blackjack, at least
when

trying to detect *small* gains in excepted value.

What is the probability of being (720-300)/5 = 64 units ahead

after 1031 bets? Expected value is about zero for the null hypothesis

that we are no better off than using the count with a 1-4 spread.

Variance per hand is about 2.0 units squared, according to my

simulations of a 1-4 spread on an 8 deck game with AC rules. Normalizing

this we get z=(64-0)/sqrt(1031*2.0)=1.41, and looking this up

in a statistical table for the normal distribution, we get

about 8%. There is an 8% chance of results as good as mine

even with no advantage. This is not low enough that I can

disregard the possibility that my winnings could have just

been luck. In addition, the variance was probably a bit

higher than 2.0 units squared for the shuffle-tracking betting.

A Closer Look at the Bankroll Data

----------------------------------

Each trial will be now be defined as the increase in bankroll from one shoe

to the next. Since I did not vary my bets according to my bankroll

size (only according to true count and shuffle tracking), the "with"
trials

can be compared fairly to the "without" trials. We will assume that
the

changes in bankroll are drawn from a normal distribution centered at our

expected winnings. (This is possibly a shakey assumption - the distribution

of games is normal in the long run, but what about the winnings of one shoe?)

We will assume that the variance is unknown. At first I thought that

the variances should be equal (though unknown) for the "with" and

"without" trials, since our null hypothesis is going to be that shuffle

tracking doesn't do squat for our winnings, in which case one might think

it shouldn't do squat to the variance either. However, the shuffle tracking

did allow me to increase my bet size more often; therefore the variance

should be higher in the "with" trials. Given that the null hypothesis

is that the means (but not necessarily variances) are equal, the

alternative hypothesis is that shuffle tracking gives us a higher

average win per shoe.

Here is the data for per shoe wins without tracking:

0.00 +40.00 -20.00 +27.50 +30.00 +5.00 +7.50 -2.50 -15.00

+30.00 0.00 +5.00 +5.00 -2.50 +2.50 -45.00 -25.00 -15.00

+7.50 -12.50 -7.50 +22.50 -7.50 +5.00 -7.50 +10.00 -55.00

+7.50 +7.50

Where is the data for per shoe wins with tracking:

+47.50 -32.50 +5.00 -20.00 +75.00 -32.50 +52.50 +12.50 +7.50

+12.50 -2.50 -2.50 +10.00 -37.50 -50.00 -17.50 -5.00 +90.00

-7.50 -30.00 0.00 +7.50 +82.50 +100.00 +50.00 -20.00 +2.50

+10.00 -17.50 -20.00 +2.50 +160.00 -12.50

Let u1 be the actual average shoe win "without tracking"

Let u2 be the actual average shoe win "with tracking"

Let x1 be the mean of the observed shoe wins "without tracking"

Let x2 be the mean of the observed shoe wins "with tracking"

Let n1 be the number of observed shoes "without tracking"

Let n2 be the number of observed shoes "with tracking"

Let S1^2 be the sample variance of the observed shoe wins "without tracking"

Let S2^2 be the sample variance of the observed shoe wins "with tracking"

Then

H0: u1 = u2

H1: u2 > u1

u1 = ?

u2 = ?

x1 = -0.086207

x2 = +12.727

n1 = 29

n2 = 33

S1^2 = 412.277

S2^2 = 2058.85

We should reject the null hypothesis if |t0*| > t(alpha, v) and t0* negative

where in LISPish notation,

t0* is (/ (- x1 x2) (sqrt (+ (/ S1^2 n1) (/ S2^2 n2)))

and v is (- (/ (sqr (+ (/ S1^2 n1) (/ S2^2 n2)))

(+ (/ (sqr (/ S1^2 n1)) (+ n1 1))

(/ (sqr (/ S2^2 n2)) (+ n2 1))))

2)

(Hairy formulas courtesy of "Probability and Statistics in Engineering

and Management Science" by Hines and Montgomery)

Plugging in we get...

t0* = -1.46395

v = 46.41

Looking in the t distribution table, t(.1, 40)=1.3 while t(.05,40)=1.7

so we can reject the null hypothesis, but with only 90% confidence.

So again there is that ~10% that I cannot ignore.

APPENDIX III

Computer Trial Shuffle-Tracking Empirical Results

[The computer was found to be using a slightly incorrect shuffle-tracking

profile after this report was written. Once this was fixed, the

shuffle-tracking did even better, cutting at a true count of +7.00

count instead of +6.45, and had about error rate about 1.5% lower.]

The results are astounding and clear: my shuffle-tracking procedure

is clearly better for sizing one's bet than using true count given the

Random Pick Order Six Zone Shuffle. This contradicts Patterson's wisdom,

expressed in "Break the Dealer", that one should use shuffle-tracking
for

cutting the cards, but not for sizing one's bets; I should perhaps state

that I've never had much faith in Patterson's assertions, and what I know of

his TARGET system doesn't impress me - sorry Jerry. My shuffle-tracking

procedure allows one, given the cut card, to remove an average count of over

+6 from the played cards (the first 5 1/3 decks). This in itself

neutralizes the base casino advantage with Atlantic City rules, even

for basic strategy players unknowingly at the same table as the

tracker. However, in addition, the shuffle-tracking estimates of

advantage are more accurate than the true count.

I used shuffle-tracking units of 2/3's decks, and my shuffle-tracking

procedure calls for just 6 additions of 2 numbers during the shuffle.

This is an approximation for simplicity, but it is fairly accurate.

If I had the computer use the precisely correct shuffle-tracking

procedure and had the simulated dealer perform perfect pick sizes,

then the computer would predict exactly the count of the next region;

however, this would not have been very informative. Instead I had the

computer simulate my human-manageable shuffle-tracking procedure and a

nonperfect casino shuffle. I tried to make the shuffle as realistic as

possible; it is for the Claridge's eight deck "zone" shuffle. All
the

simulation's picks were subject to a +-5% error, and the simulation's

riffle shuffles were imperfect. This simulation does not actually

play blackjack; it just flips over the cards and places them in the

discard tray one at a time, but this allows us to see how accurate

tracking is in predicting when the high cards are going to come out.

I ran the simulation for 100,000 shoes (starting with a totally

pseudorandomly shuffled shoe), so the results are very accurate (to maybe

+-0.02.) The "regions" mentioned in the data summary are two tracking

units, or 4/3's decks. Penetration was set at 2/3's, so only 4 of the

6 regions are dealt out, which is lousy but realistic. "ACTUAL_COUNT"

is the count for the cards in that region of the shoe. "-TRUE_COUNT"

is the negative of the running count divided by the remaining

regions, which is the true count's prediction of the next cards about

to come out; obviously the count predicts "0" for region 1 before
any

cards have been dealt. "SHUFFLE_TRACK" is the shuffle tracking

prediction, obtained by those 6 additions performed over the actual

counts for the tracking units from the previously observed shoe.

"Count cut out" is the count of the unplayed cards (the ones in

regions 5 and 6).

Here are the first five of 100,000 shoes run with random cutting:

PREDICTIONS

REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK

1 -3 0 5

2 -1 1 0

3 -5 1 -13

4 11 3 -6

Count cut out = -2

PREDICTIONS

REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK

1 1 0 14

2 2 0 2

3 -11 -1 -9

4 5 3 5

Count cut out = 3

PREDICTIONS

REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK

1 -7 0 1

2 6 1 0

3 14 0 7

4 -5 -4 -8

Count cut out = -8

PREDICTIONS

REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK

1 -17 0 -11

2 8 3 6

3 1 2 -4

4 10 3 12

Count cut out = -2

PREDICTIONS

REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK

1 -6 0 1

2 -10 1 -2

3 -1 4 -5

4 8 6 0

Count cut out = 9

If you compare "ACTUAL_COUNT" to each of "-TRUE_COUNT" and

"SHUFFLE_TRACK", then I think you can see that "SHUFFLE_TRACK"
is

better correlated to "ACTUAL_COUNT" than "-TRUE_COUNT" is,
though it

is still only a rough estimate. Since it is cutting randomly, it

cuts out an average count of 0 in the limit.

Here are the first 5 of 100,000 shoes run with intelligent cutting:

PREDICTIONS

REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK

1 7 0 0

2 -3 -1 -13

3 5 -1 -6

4 9 -3 4

Count cut out = -18

PREDICTIONS

REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK

1 -21 0 -8

2 1 4 0

3 -6 5 -10

4 -7 9 -3

Count cut out = 33

PREDICTIONS

REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK

1 -14 0 -6

2 6 3 7

3 -5 2 2

4 -9 4 -9

Count cut out = 22

PREDICTIONS

REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK

1 -5 0 -5

2 -6 1 -10

3 2 3 10

4 4 3 1

Count cut out = 5

PREDICTIONS

REGION ACTUAL_COUNT -TRUE_COUNT SHUFFLE_TRACK

1 -4 0 1

2 0 1 2

3 7 1 2

4 -9 -1 -5

Count cut out = 6

Note that it usually cuts out a positive count with the intelligent

cutting, though it messed up big time on the first shoe (I would

guess because of dealer randomness.)

If you don't believe these few trials, then here is a summary for

100,000 trials, with random cutting:

SHUFFLE TRACKING STATISTICS REPORT

PARAMETERS

----------

Number of decks: 8

Number of tracking units: 12

Number of cards per track unit: 34.666666

Number of track units dealt: 8

Number of track units per statistics region: 2

Number of trials (i.e., shoes examined): 100000

Counting system (A, 2, 3, 4, 5, 6, 7, 8, 9, 10): -1 1 1 1 1 1 0 0 0 -1

Conservative factor for shuffle tracking: 1

Type of cutting (random or intelligent): Random

+-% error in player cutting cards: 0

+-% error in dealer pick sizes: 5

% chance dealer drops 1 card in riff: 66

% chance dealer drops 2 cards in riff: 26

% chance dealer drops 3 cards in riff: 5

% chance dealer drops 4 cards in riff: 2

% chance dealer drops 5 cards in riff: 1

RESULTS

-------

Average count cut out: -0.010

IDENTIFYING FAVORABLE REGIONS

-------------------------------

CORRELATION ABSOLUTE ERROR % FALSE POS. % FALSE NEG.

--------------- -------------- -------------- --------------

REGION COUNT TRACK COUNT TRACK COUNT TRACK COUNT TRACK

------ ----- ----- ----- ----- ----- ----- ----- -----

1 N/A 0.59 5.36 4.41 N/A 33.97 41.27 25.75

2 0.19 0.59 5.21 4.36 46.53 34.01 39.16 25.52

3 0.31 0.59 5.04 4.36 42.54 33.93 35.48 25.66

4 0.45 0.60 4.79 4.39 37.55 33.55 31.95 25.77

5 (not dealt)

6 (not dealt)

OVERALL 0.29 0.59 5.10 4.38 41.10 33.87 37.46 25.67

Overall % error in identifying favorable/unfavorable regions...

COUNT: 38.10

TRACK: 28.82

Note: a region is `favorable' if and only if its count is <= -2 per region.

Okay, I've got lots of things to explain and interpret.

First, if you look up there, you'll see that the average count cut out was

close to 0, which is what we'd expect for random cutting.

Next, look at the correlation columns. The count subcolumn is the

correlation coefficient of the true count to the actual count for the

region noted to the left. The track subcolumn is the same, but for

shuffle-tracking. For the statistically underprivileged, the

correlation coefficient is a measure of the predictivity of one

statistic related to another. Correlations are always between -1.0 and

+1.0. A correlation of +1.0 means that they relate to each other

perfectly. A correlation of -1.0 means that they are exactly the

opposite of each other. A correlation of 0 means that they are

unrelated.

I can't compute the correlation for the true count for region 1,

because true count is always 0 at the start of the shoe, and this

causes the correlation equation to divide by zero. However, as one

would expect, the correlation of true count to actual count increases

as you move deeper into the shoe. In fact, for predicting the count

of the last region (6 in this case), the true count would have a

correlation of +1.0, because then you *know* the count of the

remaining region. Unfortunately, casinos realize this and thus they

don't deal anywhere near all the cards. In this experiment, the

count correlation reaches a maximum of +0.45 during the fourth

region. Compare this to tracking. With tracking, you have equal

knowledge about every part of the shoe. Thus, the tracking

correlations are all statistically indistinguishable from each other,

at about +0.59. This is considerably better than the best count

correlation, which was only obtained during the fourth region!

Overall, counting scores a +.29, while tracking scores a +.59. Hence,

the true count is only weakly correlated with the actual count, while

the tracking estimate is fairly strongly correlated with the actual

count.

Now, look at the absolute error columns. These are the averages of

the differences of the actual count with the true count and tracking

predictions. A similar pattern emerges. True count is very inaccurate

at the start of the shoe (being no help at all by guessing 0 all the

time and winding up with an error of 5.36), while deeper in the

shoe it gets better to reach a minimum error of 4.79, which is still

worse than the average error of 4.38 for the shuffle-tracking estimates.

Note that this error of 4.38 is not a huge improvement over the base

absolute error of 5.36; the shuffle-tracking technique never gets

extremely accurate in terms of predicting the absolute value of the

actual count, as a result of the simplifications in the tracking

technique and the randomness introduced into the shuffle.

You may be saying this is all well and good, but what about what

counters really care about: predicting favorable situations in

order to size one's bet. Examine the four columns on identifying

favorable regions. I defined a region as "favorable" if and only if

it contains a count of *less* than -2. For the statistically

underprivileged, false positives are when the prediction was for a

favorable region, but in actuality, the region was unfavorable; false

negatives are when the prediction was for an unfavorable region, but

in actuality, the region was favorable. Percent false positives is

the percent of the time that when the prediction is positive it is

wrong, and analogously for percent false negatives.

Since true count never predicts a favorable first region prior to

seeing it, there are no false positives, but no correct positives

either (so I cannot compute the percent false positives.) The

percent false negatives for true count in the first region therefore

happens to be the percent of actually favorable regions: about 41%.

If you look back to those sample shoes, you'll see that quite a few

shoes are "favorable" in terms of the actual count being -2 or lower.

This may be surprising, especially when you think about all the literature

that says that true count indicates an advantage less than 20% of the

time on these eight deckers, but that's just it - much of the time

you have an advantage, and much of the time that you have an

advantage you didn't predict it! However, like the previous

statistics, shuffle-tracking again does better at identifying

favorable regions in any part of the shoe than the count for the

last played part of the shoe. Overall, the true count is wrong about

the favorability of regions 38% of the time, while tracking is wrong

only 29% of the time. Just by guessing "unfavorable" all the time,

you get get an error rate of 41%, so the 38% error of true count is

pretty bad. By the way, either counting or tracking estimates can be

made more conservative in order to reduce the percent false

positives, but usually at the cost of increasing the percent false

negatives and the overall percent error.

Here's the summary of 100,000 trials with intelligent cutting.

SHUFFLE TRACKING STATISTICS REPORT

PARAMETERS

----------

Number of decks: 8

Number of tracking units: 12

Number of cards per track unit: 34.666666

Number of track units dealt: 8

Number of track units per statistics region: 2

Number of trials (i.e., shoes examined): 100000

Counting system (A, 2, 3, 4, 5, 6, 7, 8, 9, 10): -1 1 1 1 1 1 0 0 0 -1

Conservative factor for shuffle tracking: 1

Type of cutting (random or intelligent): Intelligent

+-% error in player cutting cards: 0

+-% error in dealer pick sizes: 5

% chance dealer drops 1 card in riff: 66

% chance dealer drops 2 cards in riff: 26

% chance dealer drops 3 cards in riff: 5

% chance dealer drops 4 cards in riff: 2

% chance dealer drops 5 cards in riff: 1

RESULTS

-------

Average count cut out: 6.45

IDENTIFYING FAVORABLE REGIONS

-------------------------------

CORRELATION ABSOLUTE ERROR % FALSE POS. % FALSE NEG.

--------------- -------------- -------------- --------------

REGION COUNT TRACK COUNT TRACK COUNT TRACK COUNT TRACK

------ ----- ----- ----- ----- ----- ----- ----- -----

1 N/A 0.55 5.40 4.17 N/A 26.89 58.31 37.43

2 0.21 0.56 5.26 4.47 34.48 32.57 44.06 27.48

3 0.34 0.56 5.03 4.48 34.29 34.96 39.70 25.87

4 0.51 0.54 5.31 4.23 14.00 28.44 51.88 36.11

5 (not dealt)

6 (not dealt)

OVERALL 0.32 0.57 5.25 4.34 24.66 30.27 48.71 30.93

Overall % error in identifying favorable/unfavorable regions...

COUNT: 46.66

TRACK: 30.59

Note: a region is `favorable' if and only if its count is <= -2 per region.

Okay, as you can see up there, the average count cut out was +6.45.

This in itself is reason enough to shuffle track. The other

statistics for this run are somewhat bizarre, owing partially to the

effect of this +6.45, and partially to the fact that the computer

shuffle-tracker generally thinks it has cut out more positive cards

than it actually has. From the percent false negatives for region 1,

you can see that about 58% of the regions are now favorable given

this intelligent cutting, up from 41% with random cutting. The result:

any stupid gamblers at a shuffle tracker's table may think they

are riding an incredible lucky streak, though their stupidity may

make them still lose. The count percent false positives drops from

41% with random cutting to 25% with intelligent cutting, while the

percent false negatives zooms up from 37% to 49%, pushing the overall

error from 38% to 47%, which is *worse* than the 42% error you could

get from just guessing "favorable" all the time. This is because the

count is being much too conservative in the intelligent cutting case.

Since +6.45 count is being cut out from the first 5 1/3 decks, +1.2

should be added to the true count for betting purposes given an

intelligent cut! If the true count were adjusted in this manner, then

its error in identifying the favorability of regions would drop back down.

Interestingly, the overall percent error of shuffle-tracking stays

at about 30% (though there is a statistically significant but

pragmatically insubstantial increase in its error with the intelligent

cutting.)

In summary, I assert that shuffle-tracking can kick butt over true

count in terms of sizing one's bet (and deviating intelligently from basic

strategy, as well). If you think of yourself as an expert blackjack

counter on multi-deck games but you do not shuffle-track, then

think again... you are not an expert at multi-deck blackjack unless

you shuffle-track. The above statistics have hinted at the tremendous

benefit of shuffle-tracking.

Empirical results from a full-blown blackjack simulation integrated

with the shuffle-tracking and realistic shuffles produced what seemed

to be close to a 1% boost in advantage over the non-shuffle-tracking

case. This was explained in the section "Empirical Results in

Support of Shuffle-Tracking".

In order to have 100% control of the cut card, you must take over a

table with a shuffle-tracking team, but contrary to popular belief, a

whole team is not necessary to perform the actual tracking

operations, except for complicated shuffles. Also, while having

control of the cut card is nice, one can still profit from

shuffle-tracking without having the cut card.

I would like to leave you with a word of caution. It can be difficult

to devise and implement a good shuffle-tracking scheme. Also, some

casino shuffles are much harder to track than others. If you attack

a tough shuffle or use a suboptimal shuffle-tracking scheme, you

could easily get creamed. That's why these simulation results are so

nice. I am now reasonably sure that I, armed with my shuffle-tracking

scheme, can cream this casino, rather than the other way around

(at least when the casino provides a fairly easily tracked shuffle.)

We're not talking about some measly 1.5% advantage that is the common

wisdom for the maximum advantage of counting; as Synder wrote in

the April 1990 issue of Blackjack Forum, "It is worth noting here

that a good shuffle-tracker could absolutely murder the grossly

shuffled games." Oops, I was going to leave you on a cautionary note...

how about "your mileage may vary"?!

You have permission to copy/print this report

for your own personal use only.