24 Sets, No Waiting
How many possible ways are there to set the dice? 576.
However, how many sets share the exact same SRR and SAP (Sevens Appearance Pattern)?
Yes, there are only 24 distinct sets when it comes to the SRR and SAP. This is because there are 24 ways to orient a single die, and the fact each of the 6 faces of die A match to one of 6 faces of die B to produce a 7.
Now, if we group each set of 24 "supersets" by SRR, we find some interesting things.
For instance, the hardway sets (and keep in mind I define the hardway sets as sets that show actual hardways on each primary pitch face. The "All 7" is not a "hardway" in this discussion).
The hardway sets, such as the S6, P6, 55T etc, all exist in the same superset set group. There are 24 permutations of the hardway, and they all have an identical SRR and SAP.
All 24 permutations can be had in the hardway set by rotating both dice in the same direction on the same axis at the same time by the same amount. I.e. rotate both dice 90 degrees clockwise on the Z axis, and you still have a hardway set. This holds true for all 3 axes.
With the ALL 7 set, you also have 24 permutations within the same superset. (In fact, this is true of all sets where both die have the same set of 4 faces around the same axes. I.e. die A has the 1,2,6,5 around the X axis, and die B also has 1,2,6,5 around the X axis - all 24 permutations of that set will be in the superset).
With the All 7, to make a new permutation, however, you must rotate each die in OPPOSITE directions on the same axis for the Y and Z axis (though X axis rotations are still in phase). I.e. rotate die A clockwise on Z, and rotate die B counter-clockwise on Z.
With the 1/4 yaw sets, however, it becomes more complicated. The 1/4 yaw sets are the 2V, 3V and X6.
Because of the relative orientation of the sevens in these sets, they span four separate supersets.
If you set the 3V, and rotate both die 90 degrees in the X (pitch) axes, you will get a different SRR and SAP. Let's use the 3V as an example.
With the 3V, you have four different SRRs/SAPs, depending on if the threes are on TOP, FRONT, BACK, or BOTTOM.
And here's the big tricky bit that was puzzling to me until I investigated the underlying math:
If you consider the PASS theory most critical die as being the die with the 1-6 on the Z axis faces, with the 3VL, this die is on the left, and produces a particular SRR. If you put this die on the RIGHT for the 3VR, the SRR pattern changes. UNLESS you also flop both dice upside down (180 degrees around the X axis).
So swapping the MCD from left to right also entails rotating the entire set 180 in pitch to achieve the same SRR/SAP.
This has an interesting implication for the 2V sets.
With the 2V, we have a number of interesting permutations. the 4/10 - where each primary pitch face has a 4 or 10, and the 5/9 where each primary pitch face has one of 4,5,9,10.
When you go from the 4/10 to the 5/9, the SRR MCD reverses, unless you also invert the set.
So in comparison, with the hardway set you can rotate both die the same direction and the same amount and still have the same SRR/SAP, though your box numbers will change.
The easiest one to see this on is the hardway set:
I define BACK as the faces that you see as you hold the dice to throw (front being the faces pointing forward in the direction of the throw).
So 55T33B means 55 on top and 33 on back (and 44 on front pointing forward).
The following 24 sets comprise the hardways superset, and each shares the same SAP/SRR.
11 T 33 B
11 T 55 B
11 T 44 B
22 T 11 B
22 T 44 B
22 T 66 B
22 T 33 B
33 T 11 B
33 T 22 B
33 T 66 B
33 T 55 B
44 T 11 B
44 T 55 B
44 T 66 B
44 T 22 B
55 T 11 B
55 T 33 B
55 T 66 B
55 T 44 B
66 T 22 B
66 T 44 B
66 T 55 B
66 T 33 B
I define supersets as a group of 24 sets that share the same SAP (sevens appearance pattern), and not necessarily SRR (I've ound 2 super sets that shared the same SRR with different SAPs, and AP is really the important defining factor here).
However, with the 2V/3V/X6, any such rotation can completely change your SRR.
As an example, in each of the four permutations of the 2V/3V sets I have:
Perm 1: 6.87
Perm 2: 6.19
Perm 3: 5.79
Perm 4: 6.49
Such a great disparity makes me question if I should use these sets in casino play. My general feeling is that one should choose a set where the several variations have a close correlation. The hardway set is inherently coherent in that all perms live in the same superset. But since the 3V et al. spread across 4 supersets, I thing one should be efficient in all four supersets before using this set in actual play.
The reason I believe this is that low correlation means to me that a set is less tolerant to changes in one's throw as one will have over time, as they improve, and as they move from table to table.
Dances With Dice