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TUCoPS :: Scams :: shuffle.txt

A shuffle-tracking treatise





               The Blackjack Shuffle-Tracking Treatise
                  Copyright 1990, 1991, Michael R. Hall

              You have permission to copy/print this report
              for your own personal use only.


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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


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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.


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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.


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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.


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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.


^L
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.


^L
        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.


^L
                        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.

^L
                        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"?!



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