# 3-face subgroup of the 3x3x3 cube

__This is seemingly beyond the reach of conventional software tools to solve, so I am appealing to the seemingly large number of contributors who have developed their own bespoke utilities. I'd be grateful if a table could be posted here. In particular is the diameter of the Cayley graph for this known?__

## Comment viewing options

### Possible errata

Length Positions 0 1 1 6 2 27 3 120 4 534 5 2376 6 10572 7 47040 8 209304 9 931296

### Three Face Tree Expansion

The results are solid. To check I rewrote the code without symmetry reduction and other tricks and got the same results. The code is pretty straightforward:

// States at depth for the C3v Three Face Group -(void)test: (id)argument { NSMutableSet *allStates, *current, *next,*swap; RM_Turn gen[6] = { Rr, Rs, Ur, Us, Fr, Fs }; unsigned depth, n; NSData *state, *product; NSAutoreleasePool *threadPool; threadPool = [[NSAutoreleasePool alloc] init]; for( depth = 0 ; depth < 11 ; depth++ ) { if( depth == 0 ) //Seed the generator with the identity state { allStates = [NSMutableSet setWithObject: [self identityState]]; current = [NSMutableSet setWithCapacity: 1024]; next = [NSMutableSet setWithSet: allStates]; } else { for(state in current) //Generate the states at depth + 1 { for( n = 0 ; n < 6 ; n++ ) //apply the six turns to the parent state { product = [self productStateOfOperatorState: [faceTurns objectAtIndex: gen[n]] andState: state]; if( [allStates member: product] == nil ) //test for duplicates { [next addObject: product]; [allStates addObject: product]; } } } } swap = current; current = next; next = swap; [next removeAllObjects]; [self report: [NSString stringWithFormat: @"\n%2d%10d%10d", depth, [current count], [allStates count]]]; } [[NSNotificationCenter defaultCenter] postNotificationName: RM_PROCESS_COMPLETE object: nil]; [threadPool release]; }

Output:

Depth States Total 0 1 1 1 6 7 2 27 34 3 120 154 4 534 688 5 2376 3064 6 10560 13624 7 46920 60544 8 208296 268840 9 923586 1192426 10 4091739 5284165 Job complete: time: 00:00:26.8 Ready

### You're right - my table is wrong!

Even the length 9 one did not come down by any. I suspect the transition from infinite to finite will be exact (i.e. it will be unlikely to come up with a finite proper supergroup based on the random but symmetric relations that I used).

As an exercise I also did similar investigations into presentations of infinite supergroups of two orthogonal faces, e.g. {U,F}. The divergence seems to occur at length 10 where the number of position is 11478 (30 more than in the finite 2-generator case).

BTW what are the antipodes specifically in terms of U and F in the 2-face case if I may ask? In a previous thread I see this was discussed but no mention of how many at the diameter 25q? Although I am more intersted in the 3-generator case...

### C2v Two Face Group Antipodes

As it turns out at one time I played around with the C2v two face group and happened to have saved the antipodal positions (reduced by symmetry ). Here are solutions for representative members of the twenty-seven depth 25 equivalence classes for the RU group.

Solving Cube State 1 Configuration: DR UR UF BR DF UL DB DL UB FL FR BL RBU FUL UBL RDB DRF DFL DLB FRU Symmetries: 1 x, y, z E Solution found: 25 q-turns R' U R U2 R U' R' U R' U' R' U R U' R2 U2 R U' R U' R U' Solving Cube State 2 Configuration: UF UR UB UL DF DR DB DL BR FL FR BL FDR BRD LFU LUB FRU DFL DLB BUR Symmetries: 1 x, y, z E 2 x, y,-z Sigma_h Solution found: 25 q-turns R U2 R U R' U' R' U' R2 U R U2 R2 U R U' R' U2 R' U' Solving Cube State 3 Configuration: UF DR UL FR DF UB DB DL BR FL UR BL URB RUF DRF BRD BLU DFL DLB FUL Symmetries: 1 x, y, z E Solution found: 25 q-turns U R U2 R' U R' U R U2 R U R' U' R U2 R' U R' U R' U' R Solving Cube State 4 Configuration: DR UR FR BR DF UL DB DL UF FL UB BL BLU FUL URB RDB RUF DFL DLB FDR Symmetries: 1 x, y, z E Solution found: 25 q-turns U2 R2 U2 R' U' R' U R' U R' U R U' R U R' U2 R' U' R U' Solving Cube State 5 Configuration: BR UR FR DR DF UL DB DL UB FL UF BL BUR FRU FUL BLU DBR DFL DLB DRF Symmetries: 1 x, y, z E 2 x, y,-z Sigma_h Solution found: 25 q-turns R' U2 R U R2 U R U' R U2 R U R' U R2 U' R U2 R2 Solving Cube State 6 Configuration: UB DR UF UL DF UR DB DL BR FL FR BL RBU RUF LFU LUB DBR DFL DLB DRF Symmetries: 1 x, y, z E 2 x, y,-z Sigma_h Solution found: 25 q-turns U R U R' U' R U2 R' U2 R' U R2 U2 R' U R' U R' U R U' Solving Cube State 7 Configuration: UF BR DR UB DF UR DB DL UL FL FR BL RBU UFR FDR BRD UBL DFL DLB FUL Symmetries: 1 x, y, z E Solution found: 25 q-turns U2 R' U2 R' U' R' U R' U R U' R' U R U' R U R' U R U' R2 Solving Cube State 8 Configuration: UL UR FR DR DF BR DB DL UF FL UB BL RBU DRF UFR ULF LUB DFL DLB DBR Symmetries: 1 x, y, z E Solution found: 25 q-turns R U2 R2 U R' U R U2 R U' R U2 R2 U' R2 U' R U R Solving Cube State 9 Configuration: UF BR DR FR DF UB DB DL UL FL UR BL LFU LUB RBU RUF DBR DFL DLB DRF Symmetries: 1 x, y, z E Solution found: 25 q-turns U' R U2 R' U' R' U R U R' U' R U2 R' U' R U' R2 U2 R U Solving Cube State 10 Configuration: FR UR BR DR DF UL DB DL UF FL UB BL RBU RUF ULF UBL BRD DFL DLB FDR Symmetries: 1 x, y, z E 2 y, x,-z C2 3 y, x, z Sigma_d 4 x, y,-z Sigma_h Solution found: 25 q-turns U R U R U2 R2 U' R U' R U' R2 U2 R U R U2 R U' R' Solving Cube State 11 Configuration: DR UR FR BR DF UB DB DL UL FL UF BL RBU RUF DRF RDB LUB DFL DLB FUL Symmetries: 1 x, y, z E 2 y, x, z Sigma_d Solution found: 25 q-turns U R U R U R2 U R U' R U' R U R U' R U' R U R' U R' U R Solving Cube State 12 Configuration: UF FR DR UR DF UL DB DL BR FL UB BL RFD BRD ULF UBL FRU DFL DLB URB Symmetries: 1 x, y, z E Solution found: 25 q-turns R U2 R U' R' U R' U' R' U R2 U R' U R U' R U R U' R' U R' Solving Cube State 13 Configuration: UB DR UF UL DF UR DB DL BR FL FR BL FUL BLU BUR FRU BRD DFL DLB FDR Symmetries: 1 x, y, z E 2 x, y,-z Sigma_h Solution found: 25 q-turns R' U' R U R U2 R2 U' R U R' U' R' U' R U R U R U R' U R' Solving Cube State 14 Configuration: BR UL FR DR DF UR DB DL UF FL UB BL RBU UBL BRD FDR FRU DFL DLB LFU Symmetries: 1 x, y, z E Solution found: 25 q-turns R U2 R' U' R2 U R' U R U R' U' R2 U2 R' U' R' U2 R2 Solving Cube State 15 Configuration: UF UL BR DR DF UB DB DL UR FL FR BL FDR DBR ULF UBL RUF DFL DLB BUR Symmetries: 1 x, y, z E Solution found: 25 q-turns U2 R U R' U2 R' U' R U' R2 U R' U' R2 U2 R' U R' U' R Solving Cube State 16 Configuration: BR UR UF UL DF DR DB DL UB FL FR BL FRU RBU FUL RDB LUB DFL DLB RFD Symmetries: 1 x, y, z E Solution found: 25 q-turns R U R U2 R' U R U' R U2 R2 U R U R2 U' R U R' U R Solving Cube State 17 Configuration: BR FR UR DR DF UL DB DL UF FL UB BL ULF FRU BUR UBL DRF DFL DLB DBR Symmetries: 1 x, y, z E Solution found: 25 q-turns R' U2 R' U R U2 R' U R U' R U' R U' R' U2 R' U' R' U R U Solving Cube State 18 Configuration: DR UL FR BR DF UR DB DL UB FL UF BL FDR RUF URB RDB FUL DFL DLB BLU Symmetries: 1 x, y, z E Solution found: 25 q-turns U R U2 R' U R U' R U R' U' R U' R U R' U R' U2 R' U R U Solving Cube State 19 Configuration: UF UR DR BR DF UL DB DL FR FL UB BL BRD RUF ULF FDR URB DFL DLB BLU Symmetries: 1 x, y, z E Solution found: 25 q-turns U R U R U R2 U R' U R' U' R U' R U2 R' U R U' R' U' R' U' Solving Cube State 20 Configuration: UF UR DR BR DF UL DB DL FR FL UB BL RBU ULF FRU FDR BRD DFL DLB BLU Symmetries: 1 x, y, z E Solution found: 25 q-turns R U R' U2 R' U2 R' U' R' U' R U' R U R' U' R2 U2 R U' R Solving Cube State 21 Configuration: BR UR FR DR DF UL DB DL UB FL UF BL RBU RUF FDR DBR BLU DFL DLB ULF Symmetries: 1 x, y, z E 2 y, x, z Sigma_d Solution found: 25 q-turns U R U R U2 R U2 R' U R' U R U' R U' R' U R' U' R U2 R Solving Cube State 22 Configuration: BR FR UR DR DF UL DB DL UF FL UB BL DRF UFR UBL ULF BRD DFL DLB BUR Symmetries: 1 x, y, z E Solution found: 25 q-turns U2 R' U' R2 U' R U2 R' U' R U' R U2 R2 U' R U' R' U' R Solving Cube State 23 Configuration: DR UR FR BR DF UB DB DL UL FL UF BL RBU RUF FDR RDB LUB DFL DLB ULF Symmetries: 1 x, y, z E 2 y, x, z Sigma_d Solution found: 25 q-turns U R U R U2 R' U R' U' R U R' U' R' U2 R' U R2 U' R' U' R Solving Cube State 24 Configuration: DR FR UR BR DF UL DB DL UB FL UF BL RUF RBU RFD BLU DBR DFL DLB LFU Symmetries: 1 x, y, z E Solution found: 25 q-turns R U2 R' U' R2 U' R2 U R' U' R' U R U R' U' R2 U2 R2 Solving Cube State 25 Configuration: UB DR UF UL DF UR DB DL BR FL FR BL RFD RDB ULF UBL RUF DFL DLB RBU Symmetries: 1 x, y, z E 2 x, y,-z Sigma_h Solution found: 25 q-turns U2 R2 U R' U2 R' U R' U R' U R2 U' R U R' U' R U' R' U Solving Cube State 26 Configuration: BR DR FR UL DF UR DB DL UB FL UF BL URB UFR FUL BLU DBR DFL DLB DRF Symmetries: 1 x, y, z E 2 x, y,-z Sigma_h Solution found: 25 q-turns U2 R' U2 R U' R2 U' R U2 R U2 R U2 R U' R U2 R' Solving Cube State 27 Configuration: DR UR FR BR DF UL DB DL UF FL UB BL FDR LUB ULF RDB BUR DFL DLB FRU Symmetries: 1 x, y, z E Solution found: 25 q-turns R' U R U2 R' U2 R U' R U' R' U R2 U R U' R U' R' U' R U'

### The second of these potential

What about the position (3 by symmetry) where the three faces are rotated clockwise with the exception of the three centre spots and one diagonally opposed corner to the 2x2x2 fixed block which stays fixed - together with the "mini-superflip" of the second type above?

I know the 4-spot+superflip case of M. Reid in the 3x3x3 is quite a different propostion.

I don't have a method for finding such sequences.

### Three Face Group Distribution

Your candidate position doesn't pan out. It may be solved in 22 q-turns:

Solving state: RF FU LF DF RB UB DB DL UR DR UL BL UFR FUL FLD FDR BUR BRD DLB BLU Target Depth: 0 2 4 6 8 10 12 14 16 18 20 22 Time: 00:00:00.6 U' F' R' F R2 F2 R U F U' F' U R F U' R U' R F' R

As an educated guess, I would say the diameter of the group is almost certainly not 26. I would say 27 or perhaps as high as 28. Solving 1000 random cubes gives the distribution:

Depth Count Fraction Log(n) 17 1 0.10% 11.23 18 6 0.60% 12.01 19 16 1.60% 12.44 20 46 4.60% 12.89 21 186 18.60% 13.50 22 386 38.60% 13.82 23 296 29.60% 13.70 24 62 6.20% 13.02 25 1 0.10% 11.23

The distribution maxes out at 22 turns with nearly 10^{14} states. From depth 24 to depth 25 the number of states drops by approximately two orders of magnitude. These distributions drop off in an accelerated manner on the far side of the maximum so I would expect the count at depth 26 to be four or five orders of magnitude less than the depth 25 count. This would give 10^{6} or 10^{7} depth 26 positions. From there I would expect a handful of positions at depth 27 perhaps then tailing off to a position or two at depth 28.

### Just to clarify...

I wonder if I may try your patience one last time? Could you minus out the 8-edge mini-superflip in the example above and use your software to derive the length of this unique position please?

### Three Face (almost) three spot position

Solving state: RU RF RD RB LU LF DB DL UF UB DF BL UFR RFD RDB RBU LFU LUB DLB LDF Target Depth: 0 2 4 6 8 10 12 14 16 18 20 Time: 00:00:00.0 R U R' F2 R F R F' U' R' F U' R' F R' U F' U' R'

Looking for antipodal positions is worse than looking for a needle in a haystack. I am mystified that the (probably) antipodal 26 turn 6 face position was found so easily. It does show some symmetry but I believe all the symmetrical 6 face positions have now been solved without finding a comparable position.

### Deeper Table

Using C_{3v} + inverse symmetry reduction I can extend the above table out to depth 12:

C_{3v}Three Face States at Depth Depth Classes+ Elements 0 1 1 1 1 6 2 4 27 3 12 120 4 51 534 5 207 2376 6 909 10560 7 3950 46920 8 17493 208296 9 77153 923586 10 341597 4091739 11 1510525 18115506 12 6682605 80156049

### I think the largest analysis

I think the largest analysis of this type that's been done for Rubik's cube is reducing an arbitrary cube state to the square's group (in other words, the first three Thistlethwaite phases in one step). This was done for the face-turn metric using a supercomputing cluster or something like that. It has |G| / 663,552 = 65,182,537,728,000 positions. These positions can be reduced by 48x symmetry to something on the order 1.36 trillion positions.

<U, R, F>, on the other hand, has over 2.5 times as many actual positions, and can only be reduced by about 12x using symmetry and antisymmetry. This comes to about 14.2 trillion positions. So this is basically about 10.5 times bigger than the other analysis I mentioned above.

## Three face group

Here are the counts out to depth 9:

As far as I know the diameter of the Cayley graph is unknown. It is a least 26. The two states (and their symmetry conjugates) which are as close as one can get to superflip in this group both are depth 26 in the q-turn metric.

F U R' F R U' R' U R' F R' U' U' R' F' R F' R U F' R F U' U' R' R'

F' U' R U' R' F' U' U' R' R' F' U' R' U R' F' U' U' R' U' U' R' R' U' R' R'