Five generator group of the 3x3x3 cube

The search for solutions to the more symmetrical hermit position in the 5-face metric is complete. No depth 26q solutions were found, confirming Bruce Norskog's determination. The diameter of the cube in the 5-face metric is now known to be at least 28q.

Solve: BU LU FU RU BD LD FD RD LB RB LF RF UBL ULF UFR URB DLB DBR DRF DFL
Target Depth: 26 28 
R2 U R2 U F' B' U L' F R' B L' U' F R' L' B' U L B R' L' B' L F' B (28)
Time: 57:44:32.7

And here is a 28q solution beginning with U.

U F U' B' L' R B' U L' B U L' U L' R B U' L F' L B F' R U L' U' R2

These two solutions establish the position as a local maximum—any initial turn takes the position closer to identity. (Under C4v symmetry all q-turns of the R F L B faces are symmetry equivalent and U is symmetry equivalent to U', i.e any solution with initial turn U will have a symmetry conjugate with initial U' , etc).

The two 26q 6-face positions with the 4-fold axis orthogonal to the 4-fold axis of the 5-face turn set may be solved in 26q in the 5-face metric:

Solve: BD RD FD LD BU RU FU LU RB LB RF LF DBR DRF DFL DLB URB UBL ULF UFR
Target Depth: 26 
R2 U L2 B R' U L' F' B U B' R U' R L' F L F' B' L F U F B (26)
Time: 00:25:32.2

Yes, I have many 26q solutions for the "orthogonal" case. It's the same axis case that I believe there is no 26q solution for. I suppose I should have said "one and only one of the three 26q* positions" in my earlier post.

The other 26q 6-face position may be solved in 28q in the five-face metric:

Solve: BU LU FU RU BD LD FD RD LB RB LF RF UBL ULF UFR URB DLB DBR DRF DFL
Target Depth: 28
R2 U R2 U F' B' U L' F R' B L' U' F R' L' B' U L B R' L' B' L F' B (28)
Time: 00:02:43.7

The first six turns above are on the first branch of the search tree, so I lucked out there. I'm working on the 26q depth. At the rate it's proceeding it will take several days to demonstrate the position cannot be solved in 26q

Thank you for confirming at least that this position is not > 28q for the 5-face Cayley graph.

Awhile ago, I believe I found all essentially distinct 26q* maneuvers for the known 26q* position (specifically using the case where the position is invariant under conjugation by horizontal cube rotations). "Grep'ing" through these positions, I found none that don't require moving both U and D face layers. So I believe your 26q search for this position will fail. However, I would appreciate it if you would continue the search to completion, if you feel that's reasonably possible, as an independent confirmation of my finding.

I agree it may be possible for distance-30 positions in the 5-face Cayley graph to exist, although I remain skeptical. Superflip must be at least 26q and is certainly worth checking out. To do it quicker, one could run superflip composed with the following sequences (assuming D is the disallowed face): UU, UF, UF', FU, FU', FF, FR, FR', FB, FB' (doing as many of these in parallel as possible). EDIT: I think maybe FL and FL' are also needed. But we can also stop as soon as a 26q superflip maneuver is found (24q with the 2 "premoves").

It might also be helpful to have a distance distribution of 1000 or more random positions. I'm guessing the distance distribution peaks at 23, and that would mean 30 is 7 beyond the peak. While in the 6-face Cayley graph, it appears nearly all are within 3 of the peak (21), and it appears likely that all are witin 5 of the peak.

Superflip is at depth 26q in the 5-face metric:

R U R U F B R' U' B' L' F L' F R B' U' L' U2 R' L B' L F' L U

I've solved a set of 100 random cubes:

100 States solved, average: 22.62
Depth Count
  20    3
  21    9
  22   29
  23   41
  24   18

You're spot on with your speculation that 23 is the most probable depth. I'm still searching depth 26 for the more symmetrical hermit position. I'm a bit over half way there. Should be able to report on it tomorrow afternoon.

Although the sample size is small, the distribution is looking rather similar to the regular QTM distribution offset by 2q. With the 24q's well below the 22q's along with superflip being just 26q* for this Cayley graph, I would say the prospects of a 30q* (or even 29q*) are becoming more remote.

Here are the results found from solving 1000 random cubes (actually 500 odd turn parity + 500 even turn parity random cubes.)

1000 States solved, average: 22.74
Depth Count
  20   11
  21   64
  22  305
  23  423
  24  184
  25   13

Depth 25 states are about twenty times more likely than Depth 23 states in the 6-face metric. With a branching factor of ~7.6 compared to ~9.4 for the 6-face metric, the 5-face distribution would be expected to be a bit broader and drop off less quickly. Still I would agree that a 29q diameter is rather unlikely.

I can confirm that 13f, 17q is the minimum length of the 6th generator turn sequence:

Solve: UF UR UB UL DR DB DL DF FR FL BR BL UFR URB UBL ULF DBR DRF DFL DLB
Target Depth: 3 5 7 9 11 13 15 17 
R U' F' R L' U B R' F L' B U R' L F' U' L (17)
Time: 00:00:04.3

Solve: UF UR UB UL DR DB DL DF FR FL BR BL UFR URB UBL ULF DBR DRF DFL DLB
Target Depth: 2 3 4 5 6 7 8 9 10 11 12 13 
R L' F2 B2 R L' U R L' F2 B2 R L' (13)
Time: 00:00:01.4

I let my solver crank overnight looking for 28q 5-generator solutions to the two non-equivalent "hermit" 26q 6-generator positions with no success. Even after eight hours the solver had progressed an insignificant way into the search so this doesn't mean anything.

The one I'm aware of is (performs "D" without moving D layer):

R L F2 B2 L' R' U R L B2 F2 L' R' (13f, 17q)

The first 6f/8q moves put the D layer corners and edges onto the U layer. After turning the U layer, the inverse of the first 6f/8q moves is done to put the pieces back to the D layer.