# Optimal supercube cross



## cuBerBruce (Nov 25, 2014)

I've done an analysis of the optimal number of moves (face turns) to solve each case of the supercube cross. By supercube cross, I mean solving the 4 cross edges and orienting the 5 centers adjacent to those edge positions. This is for solving a particular color cross. (It's not for color neutral solving.) The distribution is:


```
moves   positions
  -----   ---------
    0             1
    1            15
    2           158
    3          1682
    4         17469
    5        166685
    6       1425198
    7      10144474
    8      49800450
    9     104027538
   10      28994240
   11         64004
   12             6
          ---------
 total    194641920
```

The 6 antipodes include two equivalence classes. Scrambles to generate cases of these two equivalence classes are:

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

(EDIT: The above are for generating antipode cases for the D cross.)

The average number of moves for solving a particular color cross is approximately 8.7636.


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## qqwref (Nov 25, 2014)

Ooh, cool. I'm kind of surprised there are positions that take 12 moves, because the maximum for normal cross is only 8 moves. Is it possible to do the same calculations for color-neutrality? How about for building a (fixed) 2x2x2 block?


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## Stefan (Nov 25, 2014)

cuBerBruce said:


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



Are these for the D cross?

On non-super cubes, they take 7 and 8 moves (for D cross).


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## cuBerBruce (Nov 25, 2014)

qqwref said:


> Ooh, cool. I'm kind of surprised there are positions that take 12 moves, because the maximum for normal cross is only 8 moves.


Well, I'll note that the percentage of 11-move supercube cases is less than the percentage of 8-move regular cube cases.



qqwref said:


> Is it possible to do the same calculations for color-neutrality?


There are a little over 2 quadrillion positions to consider for an exact color neutral calculation. A distributed effort would seem to be required. Of course, you only need to build a table for the fixed color case, and use symmetry to do 6 table lookups to get the best case cross for each position.



qqwref said:


> How about for building a (fixed) 2x2x2 block?


A fixed 2x2x2 block should be very doable.



Stefan said:


> Are these for the D cross?
> 
> On non-super cubes, they take 7 and 8 moves (for D cross).



Thanks for pointing out that omission, Stefan. Yes, the antipode scrambles are for the D cross. I'll add that to the post.


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## cuBerBruce (Nov 29, 2014)

I've now done an analysis of the supercube cross in QTM. The distance distributiont table is given below.


```
Supercube cross (QTM)

  moves   positions
  -----   ---------
    0             1
    1            10
    2            73
    3           536
    4          3922
    5         27620
    6        184728
    7       1151210
    8       6400627
    9      28690546
   10      79587153
   11      72238639
   12       6353219
   13          3632
   14             4
          ---------
 total    194641920
```
 
The 4 antipodes are in 2 equivalence classes. The essentially distinct antipodes for the D layer cross can be generated by:

R F' R R F' L B R B R B F D D
R F L D F R B D L B F D' F' D

I've also done the 2x2x2 block for the supercube in both FTM and QTM. This means the three centers that are part of the 2x2x2 block must be correctly oriented.

```
supercube 2x2x2 block: FTM  supercube 2x2x2 block: QTM

  moves   positions           moves   positions
  -----   ---------           -----   ---------
    0             1             0             1
    1             9             1             6
    2            90             2            39
    3           942             3           288
    4          9606             4          2121
    5         89330             5         14861
    6        713910             6         97460
    7       3949020             7        577222
    8       8924097             8       2718634
    9       2528145             9       7312432
   10          5010            10       5245711
           --------            11        251349
 total     16220160            12            36
                                       --------
                             total     16220160
```


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## Stefan (Nov 29, 2014)

cuBerBruce said:


> The 4 antipodes are in 2 equivalence classes. The essentially distinct antipodes for the D layer cross can be generated by:
> 
> R F' R R F' L B R B R B F D D
> R F L D F R B D L B F D' F' D



Ha, "solved" and "off by 1". I suddenly wish alg.cubing.net had a supercube mode...

The effects on the centers are:
R F' L B'
R2 F2 L2 B2


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