Nice work! These tables are very interesting.

**Bruce, could you generate for the #of moves away, all 3674160 are from any n>=4 position? That would reveal the potential for human transitions into the easier known cases.**
The table for symmetry/antisymmetry equivalence classes is:

Code:

```
moves n=1 n=2 n=3 n=4 n=5 n=6 n=8
0 0 0 0 0 0 0 1
1 0 0 0 2 0 0 0
2 0 4 0 0 0 0 0
3 3 7 0 4 0 0 0
4 7 24 7 3 0 0 0
5 24 98 23 12 0 0 0
6 93 434 72 13 0 0 0
7 511 1710 293 59 3 0 0
8 2002 6255 1073 71 9 0 0
9 4820 13353 2038 187 12 0 0
10 1642 4501 726 129 9 6 0
11 3 36 8 7 0 2 0
```

Any chance of getting diagrams, or specific cycle descriptions for those n=4,5,6 above?

.

OK, I have created a file with all the distinct cases (with respect to symmetry and antisymmetry) with 4 or more pieces solved (excluding the trivial case of all 8 solved).

I provide the cycle structure of one representative for each case. Unfortunately, the choice of representative is rather arbitrary. Cycles can be of one of two types: "oriented" cycles and "misoriented" cycles. For "oriented" cycles, the cycle length is equal to the number of cubies in the cycle. For "misoriented" cycles, the cycle length is 3 times the number of cubies in the cycle. For "misoriented" cycles, the full cycle is not included - only up to the point where the initial cubie is repeated (but with different "orientation"). An ellipsis ("...") is used to indicate the rest of the cycle is omitted.

Cycle structure file is remarkable. Great work once again, Bruce. Here is my 2-cents on how the case representations can be made less arbitrary, and maybe further reduced by 1-turn conjugation.

n = 6, 10 moves

(DFR FRD ...)(DRB BDR ...)

(DLF LFD ...)(DRB BDR ...)

**(ULB LBU ...)(DFR RDF ...)**
(DFR DRB)

(DLF RBD)

**(ULB DFR)**
n = 6, 11 moves

(DFR RBD)

(DLF DRB)

The bold ones above can be made equivalent to their respective n=6 cases, IF 1-turn conjugate moves are also allowed. It is interesting to note that the cross-cube swap (ULB DFR) can be oriented with just cube rotations, so that only 1 case is needed for all 5 of those corner 2-cycles. {(DFR DRB),(DLF RBD),

**(ULB DFR)**,(DFR RBD),(DLF DRB)} Likewise the other 3 cases (corner twisters) can be reduced to 1 - (ULB LBU ...)(DFR RDF ...)

The reductions should be even more dramatic for n=4,5. At first, I wasn't sure if a system could be adopted to (non-arbitrarily) conjugate these for the purpose of representation, but I have an idea now. With n=4 cubes there are 6 patterns for the locations of the remaining unsolved pieces.

LL = (UFR, ULF, UBL, URB)

TL = (UFR, ULF, DRF, URB)

FL = (UBL, ULF, DRF, URB)

XL = (UBL, ULF, DRF, DBR)

SL = (UFR, ULF, DBR, URB)

**ZL = (UFR, DRF, UBL, URB)**
These unsolved piece patterns can all be made equivalent to each other by some 1-2 turn conjugate, but 2-turns would make actual case recognition very confusing. So for example, with LL as the base pattern, then the TL "Tripod" cases would 1st require 2-move conjugate turns (i.e. R' U' z') to form some equivalent LL case. Too messy to make sense of. But note that the SL and ZL patterns can always be conjugated to ANY of the others with just 1-turn! And since SL is by equivalencies = ZL, ALL the n>=4 cases could be represented by forcing the unsolved pieces by at most a 1-turn conjugation, +cube rotations, to occupy ONLY the locations in ZL = (UFR, DRF, UBL, URB). Maybe this is viable.

..