Shuffle Tracking For Imbeciles - Part 1


Well-Known Member
My original posts were archived on, but now that they have removed all advantage play related material I decided to repost them here:

Re: Shuffle Tracking For Imbeciles - Part 1

Okay, so maybe the title needs some work. If I were Arnold Snyder, perhaps it would be titled "Algebraic Approximations of Normal Distributions in Single-Pass Riffle Co-Minglings and Disbursements of Cards in the game of 21."

Maybe my title isn't so bad.

I've read several "Shuffle Tracking For Dummies" type articles, but have always found the techniques to be too difficult to apply in a casino nvironment. I decided to start looking for a way to simplify and optimize the current methods of shuffle tracking in order to facilitate their use in casino play. Why should the "dummies" have all the fun? Us imbeciles wanna win besides! The following will summarize my findings on shuffle tracking - specifically what Mason Malmuth refers to as "card domination", however I have mostly heard the term "cut-off tracking" used to describe it. This type of tracking is only effective against a single pass shuffle, but variations exist that can be implemented against various shuffles. I believe the approximations in this article will help to simplify these cases as well.

Summary of Cut-off Tracking

Cut-off tracking consists of retaining the count at the end of the shoe (after the final round has been played and the cards are about to be shuffled). Assuming a balanced count is employed, the remaining unseen cards (hereafter referred to as the "cut-off slug") must therefore have a value that is equal to the running count but with the sign reversed. For example, if the running count is -7 at the end of the shoe, then the cut-off slug contains cards that will sum to +7 in order to assure a zero final count. That means the cut-off slug is made up of mostly low cards which are bad for the player.

The next step in tracking the shuffle is calculating the "average count density" of the used cards in the discard tray (hereafter "discards"). In the above example, assuming a six-deck game with five dealt, you would figure that the discarded 5 decks with a count of -7 would average a count of -7 / 5 = -1.4 per deck. If we then shuffle our cut-off slug with one of the discard decks, we would estimate a count of 7 - 1.4 = 5.6 for the new two-deck shuffled slug. We could then cut these cards to the bottom of the shoe, adjust our starting running count, and play with a significant advantage in a four-deck game. We are essentially using the cut card to "short the deck" of cards we don't want. Similarly, this method can also be used to cut good cards do the top of the shoe.

As we can see, this can be an incredibly powerful tool to use in actual casino play. Unfortunately, the computations can be a bit too clumsy for some of us to have ready when the cut card lands in front of us. Hence the need for a system that can be employed by the average imbecile.

The Approximation Formula

So isn't there an easier way to get from point A to point B? Happily, yes! Let's take a look at the formula we have already:

Shuffled slug = cut-off slug + average count density (per slug)

Average count density = discards / (number of slugs in shoe - 1)

To break this formula down, we will see that the shuffled slug (the value of the cut-off part that we are tracking AFTER the shuffle) equals the original value of the cut-off slug plus the value of the estimated average count of each slug from the discards. The average count is found by dividing the known count (the running count before the shuffle) by the number of slugs it is comprised of (number of slugs - 1). We subtract 1 because we don't want to include the cut-off slug in our division because it has it's own value already.

This is the standard formula which most of you have probably wrestled with while the dealer is shuffling and stacking away. Although it is very straightforward, the division to find the average count density can be difficult when awkward numbers are used. How many of you would have come up with +1.4 in the above example? After a few hours of casino practice, I decided that I couldn't get it. I was having problems with switching the signs as well. I would get confused with the "negative slugs are GOOD now" concept and was afraid that I would cut a bad slug to the front by accident. So I did what anyone with the mentality of a thirteen-year-old boy would do: I whined about it being "too hard" and gave up.

A few months later I sat down with Excel and used the above formula to make a spreadsheet showing different running counts for the cut-off slug and their final outcomes. I thought that having the formula with various solutions in front of me would help me to understand the concepts and perhaps memorize some of the tricky division problems. I figured that memorizing +1.4 is easier than finding 7 / 5. However, after staring at the numbers for a while, something occurred to me. Why am I going through all of this trouble? Why am I swapping signs, subtracting slugs, and dividing "average count
Densities?” If I have to figure out how many 1.5 deck slugs are in a six-deck game I'll scream! Yes, I know the answer is 4 and it's easy to remember - but when you're starting out and the dealers are using different penetration levels, it can become maddening. That's when I saw the shortcut.


Well-Known Member
Hi folks just got done reading this thread...Excellent advice by the way, for those of counters who are fairly new to the game. One more in the steps to truly mastering the game.
As a newbie, (about 8 1/2 months now) this truly went over my head. I understand it somewhat and this is my new goal to accomplish heading into this ninth month of the journey.


Well-Known Member
This article is what originally brought me to the site months and months ago. Thanks Sonny. And there is a part 2 as well. Good stuff


Well-Known Member
Bashful C. Stupid-Butt said:
I'm lost. What is shuffle tracking in it's easiest definition?

So far this site hasn;t adversely effected my winning, so I'm still listening.
Watching and mapping the shuffle so that you can follow a single card or a small packet of cards through the shuffle and identify where they end up in the post shuffle stack.



Well-Known Member
RJT said:
Watching and mapping the shuffle so that you can follow a single card or a small packet of cards through the shuffle and identify where they end up in the post shuffle stack.
And then cutting the shoe in an advantageous manner: either cutting all the good cards to the front of the shoe or to a specific location, or cutting all the bad cards into the back of the shoe.

Sonny said:
The next step will be to make a shuffle map for each dealer.
I don't see how you can make a decent map for the middle of the pile for most shuffles. All the shuffles I've watched involve two sets of riffles, which means a quarter-deck slug gets diluted into a two-deck slug in the middle of the shoe. A RC+13 13-card slug becomes an TC+2 patch, which isn't anything to write home about.

The only exceptions are the bottom and top half-decks, which only get diluted 1:4. I can only imagine trying to track these two slugs.


Here's the technique I've been thinking about, anyone is free to give me comments.

In the interest of full disclosure, I've only been able to experiment on three shoes - one without a match, one successfully, and one unsuccessfully. So who knows if this really works.

The game is a 6D, 75% penetration game. The shuffle seems to be standardized across many dealers, and involves breaking the unplayed cards into three sections and inserting them about 1/4, 1/2, and 3/4 up the played discard stack prior to shuffling. This means, specifically, that the first half-deck played is the bottom half-deck on the to-be-shuffled-pile and the last half-deck played is the top half-deck on the pile.

The key is to track 5 slugs, and to hope that they match when shuffled.

(A) First quarter-deck played.
(B) Quarter deck from 2-2.25 decks up the discard shoe during play
(C) Quarter deck from 2.25-2.5 decks up
(D) Last quarter-deck played.
(E) Unplayed cards.

The hardest part is to estimate exactly where (E) gets inserted relative to (B) and (C). This is difficult because the unplayed cards are frequently not broken into three exactly equal sections. The whole point is that you need to know what the count is like a quarter-deck above and below the 3-deck mark.

Case #1: (E) placed below (B). 3-deck mark is between (B) and (E).
Shoe, in quarter-deck segments, now looks like this
(bottom = left, top = right, x = unknown)
Axxx xEEx xxEE BCxx xEEx xxxD

Case #2: (E) splits (B) and (C). 3-deck mark is now in (E).
Axxx xEEx xxBE ECxx xEEx xxxD

Case #3: (E) placed above (C). 3-deck mark is between (E) and (C).
Axxx xEEx xxBC EExx xEEx xxxD

The first riffle is straightforward and involves cutting the pile in half, taking half-deck sections and building a new pile. A gets cut with whatever was just above the 3-deck mark, and D gets cut with whatever was just below the 3-deck mark. The order is also inverted, so that, in whole-deck segments (bottom = left, top = right), the pile now looks like this:

Case #1: (DEEx) (xxxx) (xxEE) (xxEE) (xxxx) (ABCx)
Case #2: (BDEx) (xxxx) (xxEE) (xxEE) (xxxx) (ACEx)
Case #3: (BCDx) (xxxx) (xxEE) (xxEE) (xxxx) (AEEx)

The second riffle is more complicated. Two portions are riffled, but only half is put onto the pile while the other is riffled with another portion, and thus the entire middle is diluted 1:4, in addition to being diluted 1:2 in the first riffle. As I pointed out, however, two half-decks, the very first and the very last, only get cut an additional 1:2 in this second riffle, so only suffer a 1:4 dilution overall. Now, in half-deck segments, the pile looks like this:

Case #1: [ABCEExxx] [x] [x] [x] [x] [x] [x] [x] [x] [x] [x] [DEEEExxx]
Case #2: [ACEEExxx] [x] [x] [x] [x] [x] [x] [x] [x] [x] [x] [BDEEExxx]
Case #3: [AEEEExxx] [x] [x] [x] [x] [x] [x] [x] [x] [x] [x] [BCDEExxx]

The point here is that you're hoping for a big match: DE or ABCE for case #1, BDE or ACE for case #2, or BCDE or AE for case #3. A, B, C, and D should each have a running count of their own - I find it's easiest to invert them from normal high-low so that positive numbers indicate large numbers of face cards. E is the unplayed portion - dividing the final running count by 6 will normalize the value.

The RC of the half-deck is simply the sum of all the letters in the matched slug (x = 0) divided by 8. Negative slugs are almost as valuable as positive ones, because you can cut out a negative slug.

I'm just going to use a [++] to designate a positive matched slug and a [--] to designate a negative matched slug, with [+] and [-] used for the next half-deck from the end (which is half-strength from the ends).

Front slug matches positive
[++] [+] [x] [x] [x] [x] [x] [x] [x] [x] [x] [?]
Cut anywhere behind 1.5 decks from the front (3-deck cut shown)
Played: [x] [x] [x] [x] [x] [?] [++] [+] [x] Unplayed: [x] [x] [x]

Front slug matches negative
[--] [-] [x] [x] [x] [x] [x] [x] [x] [x] [x] [?]
Cut 1 to 1.5 decks from front
Played: [x] [x] [x] [x] [x] [x] [x] [x] [?] Unplayed: [--] [-] [x]

Back slug matches positive
[?] [x] [x] [x] [x] [x] [x] [x] [x] [x] [+] [++]
Cut 1 deck from back
Played: [+] [++] [?] [x] [x] [x] [x] [x] [x] Unplayed: [x] [x] [x]

Back slug matches negative
[?] [x] [x] [x] [x] [x] [x] [x] [x] [x] [-] [--]
Cut 1 deck (or less if allowed) from the front
Played: [x] [x] [x] [x] [x] [x] [x] [x] [-] Unplayed: [--] [?] [x]