# Super Cube States at Depth

Super Cube States at Depth

I've been working with the super cube group (the 3x3x3 cube with center cubie orientation). There are two earlier threads here dealing with this group, Lower bounds for the 3x3x3 Super Group and Supergroup knowledge. Neither of these contain any states at depth information. To test my model I have performed a breadth first states at depth enumeration of the group out to depth 10 in the qtm. Can anybody confirm these numbers for me?

```Super Cube States at Depth: Max Depth 10 Turns: 12

Depth    Reduced(Oh+)      Count
0             1             1
1             1            12
2             5           114
3            17         1,068
4           130        10,011
5         1,031        93,840
6         9,393       878,880
7        86,193     8,223,552
8       803,257    76,891,875
9     7,491,172   718,649,064
10    69,967,562 6,714,751,636
Time: 45:55.66
```

## Comment viewing options

### partial confirmation

I made a quick modification to my RubikBandage program, from which I confirm the non-reduced numbers up to distance 8. I note the numbers are the same as the regular cube group to distance 6.

```\$ a.exe q
Bandaging 0 corners to edges and 0 edges to centers.
Allowed turns: U U' D D' L L' R R' F F' B B'
distance   1: count =        12, total        13
distance   2: count =       114, total       127
distance   3: count =      1068, total      1195
distance   4: count =     10011, total     11206
distance   5: count =     93840, total    105046
distance   6: count =    878880, total    983926
distance   7: count =   8223552, total   9207478
distance   8: count =  76891875, total  86099353
Hash table full! (110000550)
```

I also ran it allowing half turns.

```\$ a.exe
Bandaging 0 corners to edges and 0 edges to centers.
Allowed turns: U U' U2 D D' D2 L L' L2 R R' R2 F F' F2 B B' B2
distance   1: count =        18, total        19
distance   2: count =       243, total       262
distance   3: count =      3240, total      3502
distance   4: count =     43239, total     46741
distance   5: count =    575256, total    621997
distance   6: count =   7635214, total   8257211
distance   7: count = 101280342, total 109537553
Hash table full! (110000550)
```

### Thanks

Thanks Bruce.

The shortest actions on the centers are 14q:

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

So, the super group distribution is the same through depth 6. All states depth 6 or less are within 12 moves of one another. At depth 7 one begins to encounter configurations which differ only in the twist of the center cubies. In the ftm the shortest action on the centers is 10f. The distributions diverge at depth 5 for the ftm although I don't have the figures at hand to check that.

### got to distance 10

I implemented symmetry reduction (but not antisymmetry) in order to get to distance 10. I only used a 32-bit variable to count the total positions at each depth, so the actual distance-10 count given by the program was 232 less than shown, but obviously it had to have overflowed once, and not more than once, based upon the branching factor.

```distance  sym-reduced   sym-reduced   positions
cumulative
0             1           1             1
1             1           2            12
2             5           7           114
3            25          32          1068
4           219         251         10011
5          1978        2229         93840
6         18395       20624        878880
7        171569      192193       8223552
8       1602756     1794949      76891875
9      14974314    16769263     718649064
10     139898950   156668213    6714751636
```

EDIT: I figured I might as well run the FTM case with symmetry reduction also. Here are the results.

```distance  sym-reduced   sym-reduced   positions
cumulative
0              1           1             1
1              2           3            18
2              9          12           243
3             75          87          3240
4            934        1021         43239
5          12086       13107        575256
6         159504      172611       7635214
7        2111637     2284248     101280342
8       27979161    30263409    1342720353
9      370747712   401011121   17794854698
```