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I have computed optimal solutions for following subgroups of the cube:

Group that fixes cubies


Distance        Nr of pos       Unique wrt M    Unique wrt M + inv

0q              1               1               1
2q              0               0               0			
4q              0               0               0
6q              0               0               0
8q              0               0               0
10q             0               0               0
12q             441             11              8
14q             3944            87              52
16q             32110           708             396

In this paper we prove that Rubiks cube can be solved in at most 28 face turns. The proof uses exactly the same method as Michael Reid used in 1995. The only difference is that we show how to avoid the positions at distance 29. The proof was based on the fact that it takes 12 face turns max to take an arbitrary element of the Rubiks cube group into the group H=< U,D,L2,F2,B2,R2 >. And at most 18 face turns to take an element of H to the identity.

The above method can also be formulated in following way:

Given an arbitrary element g in the cube group we multiply it by an element B^-1 from the right such that gB^-1 is in the group H and the length of B^-1 is at most 12 face turns. Then we multiply gB^-1 by an element A^-1 from the right such that gB^-1A^-1=id --> g=AB. And the length of A is at most 18 face turns.