# Permutations of the corners and edges: FTM

I have finally completed the face-turn metric version of the analysis of the permutations of the corners and edges of the 3x3x3 Rubik's Cube. That is, I have generated a table of the Cayley graph distances for the positions where the permutation of the corner cubies and the permutation of the edge cubies are considered, but not the orientation of either the edge or corner cubies. This is a set of 8!*12!/2 or 9,656,672,256,000 positions. As with the quarter-turn metric analysis, I used symmetry in the corner permutations to reduce the number of stored distances to 984*12!/2 or 235,668,787,200.

I also plan to compute the distribution of the distances in terms of unique positions with respect to symmetries of the cube, and also the distribution where antisymmetry is also used. The reduced counts in the table below are upper bounds on the values for the number of positions that are unique with respect to symmetries of the cube.

```Permutations of the corners and edges of the
3x3x3 Rubik's Cube, ignoring orientation

Distance         Positions    "Reduced count"
--------         ---------    ---------------
0f                    1                  1
1f                   18                  2
2f                  243                  9
3f                3,240                 98
4f               43,239              1,251
5f              574,308             15,546
6f            7,599,538            198,547
7f          100,336,092          2,558,710
8f        1,320,811,484         33,223,104
9f       17,277,892,377        430,571,941
10f      220,284,799,718      5,453,945,549
11f    2,346,113,524,904     57,687,613,279
12f    6,808,973,777,807    165,731,116,833
13f      262,592,892,783      6,329,542,290
14f                  248                 40
-----------------    ---------------
9,656,672,256,000    235,668,787,200
```

There are 18 antipodes when counting positions that are unique with respect to symmetries of the cube. Move sequences for generating representatives of the antipodes are:

U D R' B' D' L2 D F B D2 F D2 L B
U D B' U2 D2 B2 R' U B U2 R2 D B' L'
U D F U' F2 L' B L' U F' D2 F' D' B2
U D F L U D2 B' D' L' B D2 F2 U' B
U D F B L2 U D R' F' B' U D F' B'
U D2 F2 U2 L U2 B' U R F2 L2 D' F U2
U D F D R B' L B U2 R F B L D'
U D F' U' R U' D B L' F D R2 L' F'
U D' R L' D2 R F' B' R2 U2 R' L' B' L2
U D R U D2 F B2 U R U2 R L U R'
U D2 B D' B2 U R L2 B R2 F2 R2 U' B
U D B2 R F' U F U F' R2 D2 F' D' R
U D2 F B U' F2 B2 D R2 L2 U' R L D'
U D R2 F' R' L F2 R2 F B R' B2 U2 R2
U D2 R' U2 F2 D R B L D' F L2 F' B2
U D B U' L D B' L' F2 L' B2 R U B2
U F U R2 F D F' L2 U2 D R2 L' U B2
U F' D' R2 U' F' D2 F2 U' L2 F2 L2 D B

- Bruce Norskog

## Comment viewing options

### Symmetry- and antisymmetry-reduced numbers

I have now calculated the number of positions that are unique with respect to the symmetries of the cube, and the number of postions that are unique with respect to the symmetries of the cube and inverses.

```Permutations of the corners and edges of the
3x3x3 Rubik's Cube, ignoring orientation

Distance       Positions      Unique wrt M  Unique wrt M+inv
--------       ---------      ------------  ----------------
0f                   1                 1                 1
1f                  18                 2                 2
2f                 243                 9                 8
3f               3,240                75                48
4f              43,239               934               509
5f             574,308            12,066             6,194
6f           7,599,538           158,743            79,988
7f         100,336,092         2,091,849         1,048,365
8f       1,320,811,484        27,521,978        13,771,573
9f      17,277,892,377       359,974,828       180,038,239
10f     220,284,799,718     4,589,335,308     2,294,966,955
11f   2,346,113,524,904    48,877,623,050    24,440,810,817
12f   6,808,973,777,807   141,854,272,251    70,934,468,707
13f     262,592,892,783     5,470,812,680     2,736,830,237
14f                 248                18                17
-----------------   ---------------   ---------------
9,656,672,256,000   201,181,803,792   100,602,021,660
```
- Bruce Norskog

### FTM/QTM Grid

I have calculated the number of positions for each combination of QTM distance and FTM distance for the permutations of the corners and edges (ignoring orientation) analysis. I have done it for overall positions, positions unique with respect to symmetries of the cube, and positions unique with respect to symmetry and inverses. While tables like this generally put FTM on one axis and QTM on the other axis, to keep the table of reasonable width, I have included a column for giving the combination of FTM and QTM distances, and the other columns providing the corresponding position counts.

```Permutations of the corners and edges of the
3x3x3 Rubik's Cube, ignoring orientation

FTM/QTM
distance     Positions  Unique wrt M  Unique wrt M+inv
--------     ---------  ------------  ----------------
0f,0q                1             1             1
1f,1q               12             1             1
1f,2q                6             1             1
2f,2q              108             4             4
2f,3q              108             3             2
2f,4q               27             2             2
3f,3q              960            22            15
3f,4q             1440            32            19
3f,5q              720            16            10
3f,6q              120             5             4
4f,4q             8544           185           109
4f,5q            17088           362           185
4f,6q            12816           277           154
4f,7q             4272            92            48
4f,8q              519            18            13
5f,5q            76032          1600           836
5f,6q           189528          3967          2015
5f,7q           189504          3966          2022
5f,8q            94224          1990          1029
5f,9q            23088           487           254
5f,10q            1932            56            38
6f,6q           675492         14130          7216
6f,7q          2017728         42090         21087
6f,8q          2518200         52553         26503
6f,9q          1664160         34728         17409
6f,10q          608898         12797          6520
6f,11q          109536          2303          1166
6f,12q            5524           142            87
7f,7q          5986392        124880         62807
7f,8q         20879088        435190        217958
7f,9q         31251456        651340        326244
7f,10q        25805490        537946        269487
7f,11q        12559632        261826        131224
7f,12q         3432000         71827         36170
7f,13q          413772          8638          4357
7f,14q            8262           202           118
8f,8q         52913512       1102844        552892
8f,9q        211597104       4408868       2205254
8f,10q       370234974       7714227       3860474
8f,11q       368018376       7667952       3835193
8f,12q       224091580       4669598       2337447
8f,13q        80300580       1673244        837065
8f,14q        13333648        278521        139856
8f,15q          321672          6720          3388
8f,16q              38             4             4
9f,9q        465606300       9701400       4854820
9f,10q      2117692320      44120720      22066364
9f,11q      4295845096      89500251      44761704
9f,12q      5105816534     106375867      53200761
9f,13q      3739132476      77901404      38960088
9f,14q      1414756983      29478191      14745115
9f,15q       139015284       2896376       1449052
9f,16q           27384           619           335
10f,10q     4051435966      84408572      42220275
10f,11q    21536942020     448694176     224376182
10f,12q    53740362543    1119607519     559887452
10f,13q    78564080056    1636768834     818460665
10f,14q    52346691627    1090576272     545367765
10f,15q    10040117420     209171969     104600351
10f,16q        5170086        107966         54265
11f,11q    33548531432     698936801     349528714
11f,12q   252432622044    5259046844    2629729367
11f,13q   855359575472   17820068930    8910724709
11f,14q   920366072402   19174398757    9587899025
11f,15q   284134407796    5919497275    2960086739
11f,16q      272315746       5674442       2842262
11f,17q             12             1             1
12f,12q   195315571574    4069096245    2034734512
12f,13q  1911512441880   39823320487   19913400209
12f,14q  3206355450152   66799362228   33402523135
12f,15q  1492949068428   31103293624   15554155858
12f,16q     2841245737      59199665      29654991
12f,17q             36             2             2
13f,13q    43840406436     913357167     456875687
13f,14q   130857519042    2726244685    1363584202
13f,15q    87496405308    1822904039     912191911
13f,16q      398561824       8306780       4178428
13f,17q            172             8             8
13f,18q              1             1             1
14f,14q             12             1             1
14f,15q            184            14            13
14f,16q             52             3             3
-------------  ------------  ------------
9656672256000  201181803792  100602021660
```

### Congratulations Bruce. I have

Congratulations Bruce. I have been looking foward too see this for some time. However I was convinced that there will be more antipodes. It is also interesting to note that for your both computations the diameter was the same as for the edges only.

Yes, I was well aware that the diameters in both metrics are the same as for the edges only analysis. I guess I was a little surprised at the result that after the peak distance, how high a percentage of the remaining positions were just 1 more move deep.

I will note that none of the deep positions (17q, 18q) in QTM are antipodes in FTM. There are some corner configurations in common and at least one edge configuration in common. The following table shows these deep QTM positions and optimal sequences for them in FTM.

```                     Distance
CP          EP   QTM   FTM   FTM Move Sequence
-----   ---------   ---   ---   -----------------
54   130377481   17q   13f   U' D2 B' R  U2 R' F2 B' R  F' L2 B' U'
127   163710727   17q   13f   U  R' F  R2 U  R' D2 F' R  U2 R' U  B2
415     3653859   17q   13f   U  D' R  F  L2 D' R2 U  R' D2 F2 D  R'
4761   123758783   17q   13f   U  R  U2 D' L2 U2 F' B2 L2 D' L' U  F2
5209   127387463   17q   13f   U  B2 U2 L  B  R' L  B' R' D2 B2 D' R'
9800   127389022   17q   12f   U' L2 U2 L  U' R2 F' L2 B2 D2 F' U2
9800   127390445   17q   13f   U2 B' U  D2 F2 L  D' B2 D  F' B  R' U'
12322   299547493   17q   13f   U  F' B' U  R2 L2 U' R2 L2 D' R  L  D'
14128   409720015   17q   11f   U2 D2 R2 U' L2 F' B  U2 L  U2 F'
14128   449636809   17q   12f   U  D' L2 D2 F  B  L2 U' B2 R' L  D2
14854   248577983   17q   13f   F  U2 B' R2 F' U  B2 R' U' L  U2 D  B'
35152   127387583   18q   13f   U  D  F2 R' U2 F  B' L2 U' R  L' B2 R2
```

I have also determined that all of the antipodes in the FTM analysis have symmetry. The table below shows the corner, edge, and overall symmetries for the antipodes. It also shows the order of the positions along with the order of the corner and edge configurations of the positions.

```FTM Antipode (14f) positions in corner and edge permutations analysis

Position        Symmetries      Order
CP         EP     C   E  All    C   E All   Move sequence
-----  ---------    --   -  ---    -  -- ---   -------------
54    3662626    12   8   4     2   2   2   U D R' B' D' L2 D F B D2 F D2 L B
54  123766418    12   8   4     2   2   2   U D B' U2 D2 B2 R' U B U2 R2 D B' L'
150    3657583     4   4   4     4   2   4   U D F U' F2 L' B L' U F' D2 F' D' B2
150    3663952     4   4   4     4   2   4   U D F L U D2 B' D' L' B D2 F2 U' B
168    3645909     4   4   4     4   4   4   U D F B L2 U D R' F' B' U D F' B'
294    3653138     4   8   4     4   2   4   U D2 F2 U2 L U2 B' U R F2 L2 D' F U2
294    3658179     4   4   4     4   4   4   U D F D R B' L B U2 R F B L D'
949  299576877     6   6   6     6   6   6   U D F' U' R U' D B L' F D R2 L' F'
5176      16687     4   8   4     2   2   2   U D' R L' D2 R F' B' R2 U2 R' L' B' L2
5209  123771602     4   4   4     4   4   4   U D R U D2 F B2 U R U2 R L U R'
6068  127387583     2  48   2     6   2   6   U D2 B D' B2 U R L2 B R2 F2 R2 U' B
6754  119761927     2   4   2     8   4   8   U D B2 R F' U F U F' R2 D2 F' D' R
9682   40291210     4   8   4     8   4   8   U D2 F B U' F2 B2 D R2 L2 U' R L D'
9758  127400524     8   4   4     4   4   4   U D R2 F' R' L F2 R2 F B R' B2 U2 R2
9800  123771724     4   4   4     4   4   4   U D2 R' U2 F2 D R B L D' F L2 F' B2
14385  118721486     6   2   2     6  12  12   U D B U' L D B' L' F2 L' B2 R U B2
14385  179448349     6   6   6     6  12  12   U F U R2 F D F' L2 U2 D R2 L' U B2
35152  127427927    48   3   3     2   6   6   U F' D' R2 U' F' D2 F2 U' L2 F2 L2 D B
```

### I am probably saying the same

I am probably saying the same thing in slightly different way, but it's very interesting how short the tail of the distribution is, and how severe the drop-off is between the peak and the end of the tail.

Essentially every finite combinatorial puzzle seems to have the same sort of distribution -- a gradual rise to a peak with a relatively constant branching factor, and then the distribution has a relatively short tail after the peak.

In the case of Rubik's cube, the face turn metric seems to have a very severe droppoff and a very short tail. It seems likely to me that the diameter of the group in the face turn metric is greater than 20, but it also seems unlikely to be much more than 20.

### For me it is very interesting

For me it is very interesting to observe that it is also the number of positions distribution in % which is really almost identical for the full cube and the cube ignoring orientations when we just shift 6 moves, taking the simulation of Tom Rokicki with random cubes for15f to 20f and Jerry Bryans computation for 0f to 10f. I have no doubts, that this also holds for 11f to 14f and little doubts that this also holds for >= 20f.

 Full Cube Cube without orientations 6f 1.7*10-11% 0f 1 1.0*10-11% 7f 2.3*10-10% 1f 18 1.9*10-10% 8f 3.1*10-9% 2f 243 2.5*10-9% 9f 4.1*10-8% 3f 3,240 3.4*10-8% 10f 5.4*10-7% 4f 43,239 4.5*10-7% 11f 5f 574,308 5.9*10-6% 12f 6f 7,599,538 7.9*10-5% 13f 7f 100,336,092 0.001% 14f 8f 1,320,811,484 0.014% 15f 37 0.16% 9f 17,277,892,377 0.18% 16f 579 2.5% 10f 220,284,799,718 2.2% 17f 6,062 26% 11f 2,346,113,524,904 24% 18f 15,263 67% 12f 6,808,973,777,807 71% 19f 811 3.5% 13f 262,592,892,783 2.7% 20f 0 0% 14f 248 2.6*10-9% -------------------- 9,656,672,256,000

If this "shifting conjecture" should be true, the cube is really solvable in 20 moves. And there should be about 248*2048*2187 antipodes which means about 1 Billion 20f* cubes. Despite this number it seems very unlikely to ever get a 20f* cube by random simulation.