Double Down on Soft 12

[Gronbog]

The indices are fine, but there is a little detail that was overlooked in their generation. For example:
AAv6 assumes that two aces and a six were removed from the pack and this implies that when you receive that hand you are NOT able to split the pair of aces.
What would have to be done is to remove three aces and a six to make the calculation since the soft-12 appears after splitting the first pair.

In the mean time, until I can find time to digest what has been posted with respect to methodology, it might be reasonable to run this experiment.

Rather than removing an ace, I think the equivalent adjustment for my software would be to discard and count an ace at the start of each shoe (after the burn card). This would place all instances of A,A vs ? into a context where at least three aces have been observed and counted just they would be when playing A,A after splitting.

Does this sound reasonable?

[Cacarulo]

I store my data in a list.

If I construct a list with a specific pen and rc for the insurance count it will contain either 0 or 1 entries (elems) depending upon rc. If elems = 0, probRC = 0.

This is what I presently do:

list = new subsetList (inputCount, decks, pen, rc, specificRem);
list->getProbRC(rc, probRC, pRank);
long elems = list->elems;
delete list;
list = NULL;

What I used to do was construct a list with all rc possible to use. For more complicated counts too much data crashes program.
For the insurance count where pen = 26 there are 17 entries in the list and sum of probRC of each entry = 1. For any count sum of probRC of entries in list = 1 regardless of number of entries.

What I could do is something like this:

list = new subsetList (inputCount, decks, pen, specificRem);

for (rc = minValue; rc <= maxValue; ++rc)
     list->getProbRC(rc, probRC, pRank);

long elems = list->elems;

I can relate to probRC, but probTC? (relative to simple insurance count)

k_c

I don't understand where you're stuck. You have all the data and algorithms correct. The only thing left for you is to calculate the TC as TC = floor (RC / Cards_Left * 52).
Here it is worth a clarification: the previous formula calculates the exact TC where the remaining decks (Cards_Left / 52) are calculated exactly without rounding or truncating.
In my program I can calculate the remaining decks in various ways.

Perhaps viewing the following RC distributions as a function of Cards_Left will help:

1) HiLo

 51 |  -1 
 50 |  -2  -1   0 
 49 |  -3  -2  -1   0   1 
 48 |  -4  -3  -2  -1   0   1   2 
 47 |  -5  -4  -3  -2  -1   0   1   2   3 
 46 |  -6  -5  -4  -3  -2  -1   0   1   2   3   4 
 45 |  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5 
 44 |  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6 
 43 |  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7 
 42 | -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8 
 41 | -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9 
 40 | -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10 
 39 | -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11 
 38 | -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12 
 37 | -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13 
 36 | -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14 
 35 | -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15 
 34 | -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16 
 33 | -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17 
 32 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18 
 31 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 30 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 29 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 28 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 27 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 26 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 25 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 24 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 23 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 22 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 21 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 20 | -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 19 | -19 -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 
 18 | -18 -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18 
 17 | -17 -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17 
 16 | -16 -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16 
 15 | -15 -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15 
 14 | -14 -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14 
 13 | -13 -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12  13 
 12 | -12 -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11  12 
 11 | -11 -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10  11 
 10 | -10  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9  10 
  9 |  -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9 
  8 |  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8 
  7 |  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7 
  6 |  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6 
  5 |  -5  -4  -3  -2  -1   0   1   2   3   4   5 
  4 |  -4  -3  -2  -1   0   1   2   3   4 
  3 |  -3  -2  -1   0   1   2   3
  2 |  -2  -1   0   1   2 
  1 |  -1   0   1

2) UnbalancedTen

 51 |  -3 
 50 |  -5  -2 
 49 |  -7  -4  -1 
 48 |  -9  -6  -3   0 
 47 | -11  -8  -5  -2   1 
 46 | -13 -10  -7  -4  -1   2 
 45 | -15 -12  -9  -6  -3   0   3 
 44 | -17 -14 -11  -8  -5  -2   1   4 
 43 | -19 -16 -13 -10  -7  -4  -1   2   5 
 42 | -21 -18 -15 -12  -9  -6  -3   0   3   6 
 41 | -23 -20 -17 -14 -11  -8  -5  -2   1   4   7 
 40 | -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8 
 39 | -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9 
 38 | -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10 
 37 | -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11 
 36 | -33 -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12 
 35 | -35 -32 -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13 
 34 | -34 -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14 
 33 | -33 -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15 
 32 | -32 -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16 
 31 | -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17 
 30 | -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18 
 29 | -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19 
 28 | -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20 
 27 | -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21 
 26 | -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22 
 25 | -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23 
 24 | -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24 
 23 | -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25 
 22 | -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26 
 21 | -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27 
 20 | -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28 
 19 | -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26  29 
 18 | -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27  30 
 17 | -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28  31 
 16 | -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26  29  32 
 15 | -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27  30 
 14 | -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28 
 13 | -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26 
 12 | -12  -9  -6  -3   0   3   6   9  12  15  18  21  24 
 11 | -11  -8  -5  -2   1   4   7  10  13  16  19  22 
 10 | -10  -7  -4  -1   2   5   8  11  14  17  20 
  9 |  -9  -6  -3   0   3   6   9  12  15  18 
  8 |  -8  -5  -2   1   4   7  10  13  16 
  7 |  -7  -4  -1   2   5   8  11  14 
  6 |  -6  -3   0   3   6   9  12 
  5 |  -5  -2   1   4   7  10 
  4 |  -4  -1   2   5   8 
  3 |  -3   0   3   6 
  2 |  -2   1   4 
  1 |  -1   2

Sincerely,
Cac

[Cacarulo]

In the mean time, until I can find time to digest what has been posted with respect to methodology, it might be reasonable to run this experiment.

Rather than removing an ace, I think the equivalent adjustment for my software would be to discard and count an ace at the start of each shoe (after the burn card). This would place all instances of A,A vs ? into a context where at least three aces have been observed and counted just they would be when playing A,A after splitting.

Does this sound reasonable?

Perfectly reasonable. That is what I suggested in a previous post. I think there's a semantic issue about removing an ace, but it's exactly what you're proposing. That's why we're counting it.
In my program I don't burn any cards although I don't think that affects the results.

Sincerely,
Cac

[k_c]

I don't understand where you're stuck. You have all the data and algorithms correct. The only thing left for you is to calculate the TC as TC = floor (RC / Cards_Left * 52).
Here it is worth a clarification: the previous formula calculates the exact TC where the remaining decks (Cards_Left / 52) are calculated exactly without rounding or truncating.
In my program I can calculate the remaining decks in various ways.

Perhaps viewing the following RC distributions as a function of Cards_Left will help:


2) UnbalancedTen

 51 |  -3 
 50 |  -5  -2 
 49 |  -7  -4  -1 
 48 |  -9  -6  -3   0 
 47 | -11  -8  -5  -2   1 
 46 | -13 -10  -7  -4  -1   2 
 45 | -15 -12  -9  -6  -3   0   3 
 44 | -17 -14 -11  -8  -5  -2   1   4 
 43 | -19 -16 -13 -10  -7  -4  -1   2   5 
 42 | -21 -18 -15 -12  -9  -6  -3   0   3   6 
 41 | -23 -20 -17 -14 -11  -8  -5  -2   1   4   7 
 40 | -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8 
 39 | -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9 
 38 | -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10 
 37 | -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11 
 36 | -33 -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12 
 35 | -35 -32 -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13 
 34 | -34 -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14 
 33 | -33 -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15 
 32 | -32 -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16 
 31 | -31 -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17 
 30 | -30 -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18 
 29 | -29 -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19 
 28 | -28 -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20 
 27 | -27 -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21 
 26 | -26 -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22 
 25 | -25 -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23 
 24 | -24 -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24 
 23 | -23 -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25 
 22 | -22 -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26 
 21 | -21 -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27 
 20 | -20 -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28 
 19 | -19 -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26  29 
 18 | -18 -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27  30 
 17 | -17 -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28  31 
 16 | -16 -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26  29  32 
 15 | -15 -12  -9  -6  -3   0   3   6   9  12  15  18  21  24  27  30 
 14 | -14 -11  -8  -5  -2   1   4   7  10  13  16  19  22  25  28 
 13 | -13 -10  -7  -4  -1   2   5   8  11  14  17  20  23  26 
 12 | -12  -9  -6  -3   0   3   6   9  12  15  18  21  24 
 11 | -11  -8  -5  -2   1   4   7  10  13  16  19  22 
 10 | -10  -7  -4  -1   2   5   8  11  14  17  20 
  9 |  -9  -6  -3   0   3   6   9  12  15  18 
  8 |  -8  -5  -2   1   4   7  10  13  16 
  7 |  -7  -4  -1   2   5   8  11  14 
  6 |  -6  -3   0   3   6   9  12 
  5 |  -5  -2   1   4   7  10 
  4 |  -4  -1   2   5   8 
  3 |  -3   0   3   6 
  2 |  -2   1   4 
  1 |  -1   2

Sincerely,
Cac

I have previously never used floor funtion.

When I change output of TC using floor function and change nothing else this is output:

Count tags {-1,-1,-1,-1,-1,-1,-1,-1,-1,2}
Decks: 1
Insurance Data (without regard to hand comp)
No subgroup (removals) are defined

**** Player hand: x-x ****
Cards   RC      TC ref

48      0       0.00
47      1       1.00
46      2       2.00
45      0       0.00
44      1       1.00
43      2       2.00
42      0       0.00
41      1       1.00
40      2       2.00
39      0       0.00
38      1       1.00
37      2       2.00
36      0       0.00
35      1       1.00
34      2       3.00
33      0       0.00
32      1       1.00
31      2       3.00
30      0       0.00
29      1       1.00
28      2       3.00
27      0       0.00
26      1       2.00
25      2       4.00
24      0       0.00
23      1       2.00
22      2       4.00
21      0       0.00
20      1       2.00
19      2       5.00
18      0       0.00
17      1       3.00
16      2       6.00
15      0       0.00
14      1       3.00
13      2       8.00
12      0       0.00
11      1       4.00
10      2       10.00
9       0       0.00
8       1       6.00
7       2       14.00
6       0       0.00
5       1       10.00
4       2       26.00
3       0       0.00
2       1       26.00
1       2       104.00

How can we transition from this and simply say TC index = 0.00 in all cases?

k_c

[Cacarulo]

How can we transition from this and simply say TC index = 0.00 in all cases?

Read again the methodology that I sent you earlier. The answer is there.
Once you assemble the arrays you will see it more clearly. This is what I get from the arrays:

|       104 |  0.00871477100434657 |  2.00000000000000000 |
|        94 |  0.00000000000157696 |  1.81282495667244370 |
|        93 |  0.00000000002582721 |  1.79999999999999982 |
|        92 |  0.00000000291990962 |  1.77090909090909099 |
|        91 |  0.00000001888279004 |  1.75000000000000000 |
|        89 |  0.00000011539482804 |  1.72727272727272685 |
|        88 |  0.00000061443999346 |  1.70000000000000018 |
|        86 |  0.00000290322938725 |  1.66666666666666630 |
|        85 |  0.00000000001263757 |  1.64705882352941169 |
|        84 |  0.00001232995767432 |  1.62500000000000000 |
|        83 |  0.00000000204895869 |  1.60000000000000075 |
|        81 |  0.00004748601158011 |  1.57142857142857140 |
|        80 |  0.00000010700247691 |  1.53846153846153832 |
|        79 |  0.00000000000264831 |  1.52631578947368407 |
|        78 |  0.00016704038144613 |  1.50000000000000044 |
|        76 |  0.00000000104259939 |  1.47058823529411731 |
|        75 |  0.00000280244582382 |  1.45454545454545459 |
|        74 |  0.00000001001713137 |  1.43750000000000000 |
|        72 |  0.00054354543772864 |  1.40000000000000013 |
|        71 |  0.00000000033898419 |  1.36842105263157876 |
|        70 |  0.00000043364161695 |  1.35714285714285721 |
|        69 |  0.00004387543124798 |  1.33333333333333348 |
|        68 |  0.00000215788328434 |  1.30769230769230749 |
|        67 |  0.00000003892371046 |  1.29411764705882293 |
|        66 |  0.00000000005561459 |  1.28571428571428559 |
|        65 |  0.00169413476984208 |  1.25000000000000000 |
|        62 |  0.00000142216800880 |  1.20011666964613362 |
|        61 |  0.00003468047098998 |  1.18181818181818166 |
|        60 |  0.00000012776135551 |  1.16666666666666696 |
|        59 |  0.00044653704872974 |  1.14285714285714279 |
|        58 |  0.00000078431276579 |  1.11764705882352944 |
|        57 |  0.00011494953741622 |  1.10000000000000009 |
|        56 |  0.00002466219327307 |  1.07692334951209756 |
|        55 |  0.00000387542778391 |  1.06250000000000000 |
|        54 |  0.00000038131483604 |  1.05237716471420817 |
|        52 |  0.00557287748507175 |  1.00000000000000000 |
|        49 |  0.00000098985500915 |  0.95024721217439700 |
|        48 |  0.00006574386419129 |  0.93037593984962408 |
|        47 |  0.00024697563712821 |  0.90909115571936860 |
|        46 |  0.00000495213522480 |  0.89473684210526305 |
|        45 |  0.00091596628308714 |  0.87500000000000000 |
|        44 |  0.00017428659431586 |  0.84629621318248338 |
|        43 |  0.00002064258472653 |  0.83333333333333326 |
|        42 |  0.00000084422952156 |  0.82608695652173880 |
|        41 |  0.00355000323823723 |  0.79999999999999993 |
|        40 |  0.00000500312850994 |  0.77276324321463064 |
|        39 |  0.00053207026460195 |  0.75197575923828264 |
|        38 |  0.00004212583313604 |  0.73676600341560983 |
|        37 |  0.00235642369969060 |  0.71428571428571419 |
|        36 |  0.00001024331819497 |  0.69565217391304324 |
|        35 |  0.00021420777749855 |  0.68736137655442942 |
|        34 |  0.00160784843995766 |  0.66666534073775563 |
|        33 |  0.00119028157534114 |  0.63725719265324676 |
|        32 |  0.00079238019541892 |  0.61562675968216229 |
|        31 |  0.00053494269277241 |  0.60003142672513032 |
|        30 |  0.00058621117922467 |  0.58461036843821323 |
|        29 |  0.00024998750897103 |  0.56785934865976717 |
|        28 |  0.00002061509815479 |  0.55440322652129603 |
|        26 |  0.02831763219996664 |  0.50000000000000000 |
|        23 |  0.00017953786589658 |  0.44632471376367122 |
|        22 |  0.00114804178525575 |  0.43265146754851591 |
|        21 |  0.00209148765373934 |  0.41546099791878849 |
|        20 |  0.00447136480366181 |  0.39093339124937432 |
|        19 |  0.00056184058125404 |  0.37492946689727213 |
|        18 |  0.00515434319090128 |  0.36020785934513228 |
|        17 |  0.00627889499237681 |  0.33385621431956813 |
|        16 |  0.00331551428946029 |  0.31459958099531204 |
|        15 |  0.00164849153076835 |  0.30304918220143839 |
|        14 |  0.01180284911857132 |  0.28377075011933750 |
|        13 |  0.00831900989971149 |  0.25415565796320805 |
|        12 |  0.00444586248330500 |  0.23660929562690006 |
|        11 |  0.00491463032112485 |  0.22331489610256541 |
|        10 |  0.02168642985412893 |  0.19999999999999990 |
|         9 |  0.00644723933782312 |  0.17778520825362779 |
|         8 |  0.01422232357584844 |  0.15916454008745787 |
|         7 |  0.00944368618281479 |  0.14108362235274344 |
|         6 |  0.01711522515242123 |  0.12463070776752874 |
|         5 |  0.01605857014446175 |  0.10583558658162687 |
|         4 |  0.02216349669607403 |  0.08640204440221574 |
|         3 |  0.02408297229015970 |  0.06586101585896612 |
|         2 |  0.02559654723463824 |  0.04650772469795619 |
|         1 |  0.02940911552579521 |  0.02693192982359423 |
|        -2 |  0.04680045940086009 | -0.02585514096402246 |
|        -3 |  0.05839132204279256 | -0.04592230359816294 |
|        -4 |  0.07090358504550298 | -0.06614055023135565 |
|        -5 |  0.03502125088674130 | -0.08663693722547899 |
|        -6 |  0.04592331313814468 | -0.10415894876732154 |
|        -7 |  0.03468944744545854 | -0.12489370469198741 |
|        -8 |  0.03636714909623130 | -0.14466592458066274 |
|        -9 |  0.02420005793126845 | -0.16451936561462860 |
|       -10 |  0.02486080551606965 | -0.18353544056261387 |
|       -11 |  0.02038495612294155 | -0.20354595308019077 |
|       -12 |  0.01532869795078525 | -0.22259503687233592 |
|       -13 |  0.03894686613946247 | -0.24954723803196077 |
|       -14 |  0.00080989335537749 | -0.26795849902205870 |
|       -15 |  0.01093248186377699 | -0.27882703196169500 |
|       -16 |  0.01406786328230392 | -0.30114621791639995 |
|       -17 |  0.00481905158117618 | -0.31896326089162186 |
|       -18 |  0.01126052264892294 | -0.33420239515465161 |
|       -19 |  0.00762527797492671 | -0.35497267983234110 |
|       -20 |  0.00537093486559662 | -0.37316482762760189 |
|       -21 |  0.01801042182419816 | -0.39999853302049365 |
|       -22 |  0.00129906745042194 | -0.42137186852973024 |
|       -23 |  0.00396274715883349 | -0.43423415635535390 |
|       -24 |  0.00524302171262204 | -0.45346298679295921 |
|       -25 |  0.00289832384058324 | -0.47175178783928612 |
|       -26 |  0.01044042271488793 | -0.49957257348688450 |
|       -27 |  0.00022173003543806 | -0.50606119544374928 |
|       -28 |  0.00357199862289537 | -0.53351642333846871 |
|       -29 |  0.00079307750523909 | -0.55079027561422045 |
|       -30 |  0.00644165281717133 | -0.57141013454565792 |
|       -31 |  0.00032944637150629 | -0.59028248638169123 |
|       -32 |  0.00130362128163072 | -0.60131673732001456 |
|       -33 |  0.00418993054904777 | -0.62500000000000011 |
|       -34 |  0.00062257914666534 | -0.64668475755092125 |
|       -35 |  0.00278975680418676 | -0.66666577754524736 |
|       -36 |  0.00023555528866450 | -0.68407953715517900 |
|       -37 |  0.00187454179945584 | -0.70000491647215080 |
|       -38 |  0.00134223589377089 | -0.72646282678620899 |
|       -39 |  0.00086745082199328 | -0.74965167535242383 |
|       -40 |  0.00056156501838238 | -0.76910060208855813 |
|       -41 |  0.00036102366584841 | -0.78567292568131819 |
|       -42 |  0.00022880167207248 | -0.79999069392982158 |
|       -43 |  0.00022828057142051 | -0.81666643961004726 |
|       -44 |  0.00008039959467130 | -0.83653844120138621 |
|       -45 |  0.00002999390339164 | -0.85424028268551233 |
|       -46 |  0.00000425403966710 | -0.87313860252004571 |
|       -47 |  0.00000015328441153 | -0.89098134070490687 |
|       -48 |  0.00000000042684590 | -0.90693430656934315 |
|       -52 |  0.04034992234489480 | -1.00000000000000000 |

Note that there is no TC floored equal to zero or equal to minus one. The index is clearly +1 since from +1 the expected value is positive.

+----------+-------+-----+-----+------------+------------+----------------------+----------------------+
|   Play   | Decks |  CR | IRC |  TC Index  |  RC Index  | Total EV >= TC Index | Total EV >= RC Index |
+----------+-------+-----+-----+------------+------------+----------------------+----------------------+
|    Ins   |   1   |   0 |  -4 |         1  |      1     |  0.08558354668387291 |  0.08558183737933568 |
+----------+-------+-----+-----+------------+------------+----------------------+----------------------+

The decimal index is +1.1

Sincerely,
Cac

[k_c]

Read again the methodology that I sent you earlier. The answer is there.
Once you assemble the arrays you will see it more clearly. This is what I get from the arrays:

|       104 |  0.00871477100434657 |  2.00000000000000000 |
|        94 |  0.00000000000157696 |  1.81282495667244370 |
|        93 |  0.00000000002582721 |  1.79999999999999982 |
|        92 |  0.00000000291990962 |  1.77090909090909099 |
|        91 |  0.00000001888279004 |  1.75000000000000000 |
|        89 |  0.00000011539482804 |  1.72727272727272685 |
|        88 |  0.00000061443999346 |  1.70000000000000018 |
|        86 |  0.00000290322938725 |  1.66666666666666630 |
|        85 |  0.00000000001263757 |  1.64705882352941169 |
|        84 |  0.00001232995767432 |  1.62500000000000000 |
|        83 |  0.00000000204895869 |  1.60000000000000075 |
|        81 |  0.00004748601158011 |  1.57142857142857140 |
|        80 |  0.00000010700247691 |  1.53846153846153832 |
|        79 |  0.00000000000264831 |  1.52631578947368407 |
|        78 |  0.00016704038144613 |  1.50000000000000044 |
|        76 |  0.00000000104259939 |  1.47058823529411731 |
|        75 |  0.00000280244582382 |  1.45454545454545459 |
|        74 |  0.00000001001713137 |  1.43750000000000000 |
|        72 |  0.00054354543772864 |  1.40000000000000013 |
|        71 |  0.00000000033898419 |  1.36842105263157876 |
|        70 |  0.00000043364161695 |  1.35714285714285721 |
|        69 |  0.00004387543124798 |  1.33333333333333348 |
|        68 |  0.00000215788328434 |  1.30769230769230749 |
|        67 |  0.00000003892371046 |  1.29411764705882293 |
|        66 |  0.00000000005561459 |  1.28571428571428559 |
|        65 |  0.00169413476984208 |  1.25000000000000000 |
|        62 |  0.00000142216800880 |  1.20011666964613362 |
|        61 |  0.00003468047098998 |  1.18181818181818166 |
|        60 |  0.00000012776135551 |  1.16666666666666696 |
|        59 |  0.00044653704872974 |  1.14285714285714279 |
|        58 |  0.00000078431276579 |  1.11764705882352944 |
|        57 |  0.00011494953741622 |  1.10000000000000009 |
|        56 |  0.00002466219327307 |  1.07692334951209756 |
|        55 |  0.00000387542778391 |  1.06250000000000000 |
|        54 |  0.00000038131483604 |  1.05237716471420817 |
|        52 |  0.00557287748507175 |  1.00000000000000000 |
|        49 |  0.00000098985500915 |  0.95024721217439700 |
|        48 |  0.00006574386419129 |  0.93037593984962408 |
|        47 |  0.00024697563712821 |  0.90909115571936860 |
|        46 |  0.00000495213522480 |  0.89473684210526305 |
|        45 |  0.00091596628308714 |  0.87500000000000000 |
|        44 |  0.00017428659431586 |  0.84629621318248338 |
|        43 |  0.00002064258472653 |  0.83333333333333326 |
|        42 |  0.00000084422952156 |  0.82608695652173880 |
|        41 |  0.00355000323823723 |  0.79999999999999993 |
|        40 |  0.00000500312850994 |  0.77276324321463064 |
|        39 |  0.00053207026460195 |  0.75197575923828264 |
|        38 |  0.00004212583313604 |  0.73676600341560983 |
|        37 |  0.00235642369969060 |  0.71428571428571419 |
|        36 |  0.00001024331819497 |  0.69565217391304324 |
|        35 |  0.00021420777749855 |  0.68736137655442942 |
|        34 |  0.00160784843995766 |  0.66666534073775563 |
|        33 |  0.00119028157534114 |  0.63725719265324676 |
|        32 |  0.00079238019541892 |  0.61562675968216229 |
|        31 |  0.00053494269277241 |  0.60003142672513032 |
|        30 |  0.00058621117922467 |  0.58461036843821323 |
|        29 |  0.00024998750897103 |  0.56785934865976717 |
|        28 |  0.00002061509815479 |  0.55440322652129603 |
|        26 |  0.02831763219996664 |  0.50000000000000000 |
|        23 |  0.00017953786589658 |  0.44632471376367122 |
|        22 |  0.00114804178525575 |  0.43265146754851591 |
|        21 |  0.00209148765373934 |  0.41546099791878849 |
|        20 |  0.00447136480366181 |  0.39093339124937432 |
|        19 |  0.00056184058125404 |  0.37492946689727213 |
|        18 |  0.00515434319090128 |  0.36020785934513228 |
|        17 |  0.00627889499237681 |  0.33385621431956813 |
|        16 |  0.00331551428946029 |  0.31459958099531204 |
|        15 |  0.00164849153076835 |  0.30304918220143839 |
|        14 |  0.01180284911857132 |  0.28377075011933750 |
|        13 |  0.00831900989971149 |  0.25415565796320805 |
|        12 |  0.00444586248330500 |  0.23660929562690006 |
|        11 |  0.00491463032112485 |  0.22331489610256541 |
|        10 |  0.02168642985412893 |  0.19999999999999990 |
|         9 |  0.00644723933782312 |  0.17778520825362779 |
|         8 |  0.01422232357584844 |  0.15916454008745787 |
|         7 |  0.00944368618281479 |  0.14108362235274344 |
|         6 |  0.01711522515242123 |  0.12463070776752874 |
|         5 |  0.01605857014446175 |  0.10583558658162687 |
|         4 |  0.02216349669607403 |  0.08640204440221574 |
|         3 |  0.02408297229015970 |  0.06586101585896612 |
|         2 |  0.02559654723463824 |  0.04650772469795619 |
|         1 |  0.02940911552579521 |  0.02693192982359423 |
|        -2 |  0.04680045940086009 | -0.02585514096402246 |
|        -3 |  0.05839132204279256 | -0.04592230359816294 |
|        -4 |  0.07090358504550298 | -0.06614055023135565 |
|        -5 |  0.03502125088674130 | -0.08663693722547899 |
|        -6 |  0.04592331313814468 | -0.10415894876732154 |
|        -7 |  0.03468944744545854 | -0.12489370469198741 |
|        -8 |  0.03636714909623130 | -0.14466592458066274 |
|        -9 |  0.02420005793126845 | -0.16451936561462860 |
|       -10 |  0.02486080551606965 | -0.18353544056261387 |
|       -11 |  0.02038495612294155 | -0.20354595308019077 |
|       -12 |  0.01532869795078525 | -0.22259503687233592 |
|       -13 |  0.03894686613946247 | -0.24954723803196077 |
|       -14 |  0.00080989335537749 | -0.26795849902205870 |
|       -15 |  0.01093248186377699 | -0.27882703196169500 |
|       -16 |  0.01406786328230392 | -0.30114621791639995 |
|       -17 |  0.00481905158117618 | -0.31896326089162186 |
|       -18 |  0.01126052264892294 | -0.33420239515465161 |
|       -19 |  0.00762527797492671 | -0.35497267983234110 |
|       -20 |  0.00537093486559662 | -0.37316482762760189 |
|       -21 |  0.01801042182419816 | -0.39999853302049365 |
|       -22 |  0.00129906745042194 | -0.42137186852973024 |
|       -23 |  0.00396274715883349 | -0.43423415635535390 |
|       -24 |  0.00524302171262204 | -0.45346298679295921 |
|       -25 |  0.00289832384058324 | -0.47175178783928612 |
|       -26 |  0.01044042271488793 | -0.49957257348688450 |
|       -27 |  0.00022173003543806 | -0.50606119544374928 |
|       -28 |  0.00357199862289537 | -0.53351642333846871 |
|       -29 |  0.00079307750523909 | -0.55079027561422045 |
|       -30 |  0.00644165281717133 | -0.57141013454565792 |
|       -31 |  0.00032944637150629 | -0.59028248638169123 |
|       -32 |  0.00130362128163072 | -0.60131673732001456 |
|       -33 |  0.00418993054904777 | -0.62500000000000011 |
|       -34 |  0.00062257914666534 | -0.64668475755092125 |
|       -35 |  0.00278975680418676 | -0.66666577754524736 |
|       -36 |  0.00023555528866450 | -0.68407953715517900 |
|       -37 |  0.00187454179945584 | -0.70000491647215080 |
|       -38 |  0.00134223589377089 | -0.72646282678620899 |
|       -39 |  0.00086745082199328 | -0.74965167535242383 |
|       -40 |  0.00056156501838238 | -0.76910060208855813 |
|       -41 |  0.00036102366584841 | -0.78567292568131819 |
|       -42 |  0.00022880167207248 | -0.79999069392982158 |
|       -43 |  0.00022828057142051 | -0.81666643961004726 |
|       -44 |  0.00008039959467130 | -0.83653844120138621 |
|       -45 |  0.00002999390339164 | -0.85424028268551233 |
|       -46 |  0.00000425403966710 | -0.87313860252004571 |
|       -47 |  0.00000015328441153 | -0.89098134070490687 |
|       -48 |  0.00000000042684590 | -0.90693430656934315 |
|       -52 |  0.04034992234489480 | -1.00000000000000000 |

Note that there is no TC floored equal to zero or equal to minus one. The index is clearly +1 since from +1 the expected value is positive.

+----------+-------+-----+-----+------------+------------+----------------------+----------------------+
|   Play   | Decks |  CR | IRC |  TC Index  |  RC Index  | Total EV >= TC Index | Total EV >= RC Index |
+----------+-------+-----+-----+------------+------------+----------------------+----------------------+
|    Ins   |   1   |   0 |  -4 |         1  |      1     |  0.08558354668387291 |  0.08558183737933568 |
+----------+-------+-----+-----+------------+------------+----------------------+----------------------+

The decimal index is +1.1

Sincerely,
Cac

OK thank you. If I'm lucky I'll be able to get it.

I still think best TC index(ins count) for +EV is greater than 0.00. I think you get >= 1.00?
> 0.00 would work for any number of decks.

k_c

[Cacarulo]

OK thank you. If I'm lucky I'll be able to get it.

I still think best TC index(ins count) for +EV is greater than 0.00. I think you get >= 1.00?
> 0.00 would work for any number of decks.

Remember that indices are generally evaluated by "greater than or equal" (>=). Also note that you could say TC >= -1 or TC >= 0 since these TCs do not exist.
The correct thing in this case is TC >= +1.

Sincerely,
Cac

[Cacarulo]

OK thank you. If I'm lucky I'll be able to get it.

I still think best TC index(ins count) for +EV is greater than 0.00. I think you get >= 1.00?
> 0.00 would work for any number of decks.

k_c

Sorry k_c, somehow a line didn't come out in my TC listing. The TC equal to zero does exist even though the expected value is zero. This is the line I accidentally leaked:

|         0 |  0.08558354668387287 |  0.00000000000000000 |

Sorry again. Anyway the index is still +1 which is where the EV starts to be positive.

Sincerely,
Cac

[Gronbog]

it might be reasonable to run this experiment.

Rather than removing an ace, I think the equivalent adjustment for my software would be to discard and count an ace at the start of each shoe (after the burn card). This would place all instances of A,A vs ? into a context where at least three aces have been observed and counted just they would be when playing A,A after splitting.

Does this sound reasonable?

Perfectly reasonable. That is what I suggested in a previous post. I think there's a semantic issue about removing an ace, but it's exactly what you're proposing. That's why we're counting it.
In my program I don't burn any cards although I don't think that affects the results.

I didn't forget about this experiment. It just took some time for me to get to it.

Under the conditions suggested above, for doubling over hitting A,A vs 3 I now get +8 as the index S17 (was +9). This matches Cac's simulation result. However, I'm still getting +8 for H17 where Cac's simulation result is +7.

Some suggestions regarding the H17 difference:

1. Encountering this hand with at least three aces and a 3 removed at +7 and +8 is a rare event occurring only 122,878 and 98,232 times respectively over the course of 100 billion rounds I simulated. As such, the EVs that were computed are separated by only 0.5 standard deviations (the standard deviation of the difference in the EVs) giving a probability of about 30.85% that doubling might be correct at +7.
2. My software never computes indices in isolation. It always includes the effects of other related index plays. It does this automatically and I've never found a practical reason to prevent it. In the case of A,A vs 3 at +8, the only relevant index play which can also be triggered downstream would be 12 vs 3 (stand) at +2 (there are others but their indices are too negative for them to be relevant in this situation). I would be interested to see what index Cac and others get for this situation (A,A vs 3, at least 3 aces removed, H17) when 12 vs 3 is also enabled. This would tend to increase the value of hitting and therefore tend to make the index for doubling higher.

[Cacarulo]

I didn't forget about this experiment. It just took some time for me to get to it.

Under the conditions suggested above, for doubling over hitting A,A vs 3 I now get +8 as the index S17 (was +9). This matches Cac's simulation result. However, I'm still getting +8 for H17 where Cac's simulation result is +7.

Some suggestions regarding the H17 difference:

1. Encountering this hand with at least three aces and a 3 removed at +7 and +8 is a rare event occurring only 122,878 and 98,232 times respectively over the course of 100 billion rounds I simulated. As such, the EVs that were computed are separated by only 0.5 standard deviations (the standard deviation of the difference in the EVs) giving a probability of about 30.85% that doubling might be correct at +7.
2. My software never computes indices in isolation. It always includes the effects of other related index plays. It does this automatically and I've never found a practical reason to prevent it. In the case of A,A vs 3 at +8, the only relevant index play which can also be triggered downstream would be 12 vs 3 (stand) at +2 (there are others but their indices are too negative for them to be relevant in this situation). I would be interested to see what index Cac and others get for this situation (A,A vs 3, at least 3 aces removed, H17) when 12 vs 3 is also enabled. This would tend to increase the value of hitting and therefore tend to make the index for doubling higher.

In the index obtained by simulation I've also included other related plays like 14v3, 13v3 and 12v3.

For S17: 14v3 = -5, 13v3 = -2 and 12v3 = +2
For H17: 14v3 = -6, 13v3 = -3 and 12v3 = +1

I think the difference we have is in 12v3. Check that index.
Note also that in these indices the difference between S17 and H17 is one less, which is also true in AA(A)v3.

Sincerely,
Cac

[k_c]

I didn't forget about this experiment. It just took some time for me to get to it.

Under the conditions suggested above, for doubling over hitting A,A vs 3 I now get +8 as the index S17 (was +9). This matches Cac's simulation result. However, I'm still getting +8 for H17 where Cac's simulation result is +7.

Some suggestions regarding the H17 difference:

1. Encountering this hand with at least three aces and a 3 removed at +7 and +8 is a rare event occurring only 122,878 and 98,232 times respectively over the course of 100 billion rounds I simulated. As such, the EVs that were computed are separated by only 0.5 standard deviations (the standard deviation of the difference in the EVs) giving a probability of about 30.85% that doubling might be correct at +7.
2. My software never computes indices in isolation. It always includes the effects of other related index plays. It does this automatically and I've never found a practical reason to prevent it. In the case of A,A vs 3 at +8, the only relevant index play which can also be triggered downstream would be 12 vs 3 (stand) at +2 (there are others but their indices are too negative for them to be relevant in this situation). I would be interested to see what index Cac and others get for this situation (A,A vs 3, at least 3 aces removed, H17) when 12 vs 3 is also enabled. This would tend to increase the value of hitting and therefore tend to make the index for doubling higher.

This is how I might handle additional removals.

//    add additional specific removals here to change pRank[]
//    pHand[0] += 1;

    subsetList *l = NULL;
    long elems;
    if (cd)
        l = new subsetList (inputCount, decks, pen, rc, pHand);
    else { // (if ! cd)
        short specificRem[10] = {0};
        l = new subsetList (inputCount, decks, pen, rc, specificRem);
    }

    l->getProbRC(rc, probRC, pRank);
    elems = l->elems;
    delete l;
    l = NULL;

//    delete additional specific removals here
//    pHand[0] -= 1;

    if (elems == 0) {
        cout << "\nAbove (player hand pen/running count/side count) input is not possible.";
        cout << "\nPress any key to continue";
        ch = getch();
        return;
    }

I guess for a sim you would remove the extra ace following reshuffle of a full shoe, then replace it before reshuffle.

For 6 decks HiLo these are my approximations for around mid-shoe (155 cards.) I just use best strategy.:

S17
AA v 3 (dbl v hit no ace removed)
RC=23 dbl: 13.514%, hit: 13.238% diff: .276%
RC=22 hit: 12.910%, dbl: 13.054% diff: -.144%
interpolate: RC=22.34, TC=7.5

AA v 3 (dbl v hit 1 ace removed)
RC=22 dbl: 13.572%, hit: 13.241% diff: .331%
RC=21 dbl: 12.972%, hit: 13.059% diff: -.087%
interpolate: RC=21.21, TC=7.1

H17
A-A v 3 (dbl v hit no ace removed)
RC=22 dbl: 13.304%, hit: 13.186% diff: .118%
RC=21 dbl: 12.699%, hit: 13.001% diff: -.302%
interpolate: RC=21.72. TC=7.3

A-A v 3 (dbl v hit 1 ace removed)
RC=21 dbl: 13.347%, hit: 13.185% diff: .162%
RC=20 dbl: 12.747%. hit: 13.003% diff: -.256%
interpolate: RC=20.61, TC=6.9

k_c

[Gronbog]

In the index obtained by simulation I've also included other related plays like 14v3, 13v3 and 12v3.

For S17: 14v3 = -5, 13v3 = -2 and 12v3 = +2
For H17: 14v3 = -6, 13v3 = -3 and 12v3 = +1

I think the difference we have is in 12v3. Check that index.

Good call to also check the index numbers above with the extra ace removed. I had not bothered to check but, with the extra ace removed, my sim agrees with all of the indices above. I think that the negative indices are too far removed from A,A vs 3 at +7 and +8 to kick in downstream and have any effect with the 4.5/6 penetration we are using.

Are you saying that your H17 index for A,A vs 3 with the extra ace removed is still +7 even with these indices enabled?

[Cacarulo]

Good call to also check the index numbers above with the extra ace removed. I had not bothered to check but, with the extra ace removed, my sim agrees with all of the indices above. I think that the negative indices are too far removed from A,A vs 3 at +7 and +8 to kick in downstream and have any effect with the 4.5/6 penetration we are using.

Are you saying that your H17 index for A,A vs 3 with the extra ace removed is still +7 even with these indices enabled?

Yes, the indices of 14v3, 13v3 and 12v3 do not vary with the extra ace removed. Therefore the index of AA(A)v3 still gives me +7.

Sincerely,
Cac