# Five generator group of the 3x3x3 cube

As is well known we can dispense with one of the Singmaster generators to still realise the whole of the 3x3x3 cube e.g. using the generating set . Apologies if this has come up before - I was wondering if there has been any analysis on the likely diameter with these five generators including inverses in the QTM? I am guessing that it will be in excess of 26.

## Comment viewing options

### Five Generator results

In the collection of God's Algorithm results that Mark Longridge collects (click the infinity graph symbol on the front page of this forum) there is the following table:

LevelNumber of PositionsLocal MaxBranching Factor
0 10
1 10010.000
2 7707.700
3 58407.584
4 4,43407.592
5 33,66407.592
6 255,32007.584
7 1,933,936...7.575
814,635,503 7.568
...... ...

Jaap
Jaap's Puzzle Page: http://www.jaapsch.net/puzzles/

### 28q?

I believe one of the three 26q* positions can not be reached in 26 quarter turns without using L layer moves. If true, that would make the diameter for this Cayley graph at least 28. Is someone able to show that this position is actually at distance 28q (disallowing L layer turns)?

A simple lower bounds calculation for the standard QTM (6-face) graph shows it requires at least 21q moves. Doing a similar calculation for the Cayley graph not allowing turns of one particular face layer, I get a result that at least 23q is necessary. Since this is two moves higher than for the 6-face graph, I would expect the antipode for the 5-face graph to be about two moves deeper than the 6-face graph. Given that 26q seems to be the likely antipode for the 6-face graph with only an extremely small number of positions more than 24q, I would guess that no more than 28q will be needed for the 5-face graph. So I'll hypothesize that 28q is the diameter of the 5-face graph.

### The 5-face diameter is 28q or greater

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

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 solution

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.

### Awesome!

I presume you're running the other one as well and hope to report on it soon?

### I am indeed

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

### Thanks

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.

### It's looking likely that one

It's looking likely that one and only one of the 26q* positions in the 6 generator group will be 28q*
in the 5 generator group Cayley graph (however that has not been confirmed yet).

However who's to say that there may not be another undiscoverd position at 30q* for for the
5 generator model? Just thought I'd thought I'd throw the cat among te pigeons!

### I agree it may be possible fo

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.

### More results

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 s

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.

### 1000 5-face random cubes

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've never seen any research

I've never seen any research on this question. The only research I'm aware of is about the question of how many moves with five generators plus inverses are required to get to the sixth generator.

Jerry

### Out of interest - sixth generator

I've not seen any reference to this question after a Google search, i.e. what is the minimal length of
the sixth generator expressed in terms of the remaining five generators?

I know there are numerous length 12 relations in the 6 generator model but these all involve more
than one repetition of each generator or its inverse so not suitable to express in terms of the 5
generator model. I am guessing that the length is nowhere near the actual (unknown) diameter
of the the 5 generator group however.

### Optimal 6th Generator Path

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 (perf

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.

### Correction: angle bracket not recognised

I meant to say in the above "using the generating set" {U,F,R,D,B,U',F',R',D',B'}.

The question is still valid.

### Input text is interpreted as

Input text is interpreted as HTML. The "angle brackets" (ASCII less than and greater than signs) are special characters in HTML. Use &lt; and &gt; to get angle brackets.

Also, if you should ever include URLs in a post, it's nice to use proper <a>-tag syntax so that they become links.