Rubik can be solved in 27f

In this paper we give a proof that Rubiks cube can be solved in 27f. The idea is to eliminate the 476 cosets at distance 12 in the group H=< U,D,L2,F2,R2,B2 >. In this way we never have to consider in the 2 phase algorithm that a coset is at distance 12. So we only solve cosets at distance 11. Together with my earlier result of 28 this gives a proof of 27. The same idea was used by Bruce Norskog in his 38q proof.

However we do not really need to compute all 476 cosets. In fact we only need to compute 7 cosets of the group T = Intersection ( < U,D,L2,F2,R2,B2 > , < F,B,L2,U2,R2,D2 > , < L,R,F2,B2,U2,D2 > ) The group H is not invariant under all symmetries. But the group T is invariant under all 48.

Let xH be one of the 476 cosets at distance 12. Then this coset can be written as a union of cosets of T. That is each xH can be written be partioned as {x1T,x2T,x3T,...}. Doing this operation for each of the 476 cosets gives us 2332400 cosets of the group T.

Now if we take one of the 2332400 cosets and conjugate it by an element m of M i.e. take xT and map it to m'xmT. If m'xmT is not in one of the 476 distance 12 cosets then we can remove it because positions from this coset can be conjugated and their conjugates can be solved with the 2-phase algorithm in max 27 moves.

I have done this operation for each of the 2332400 cosets and after conjugating each by the 48 symmetries only 40 are still at distance 12. If we reduce this 40 cosets by symmetry we are left with only 7 cosets which I solved optimally with another modification of Reids solver. As seen below all were solvable in less than 21 moves.

The first column represents the number of positions and the second the number of positions unique with respect to the subgroup of M that leaves the coset in question invariant.

BD LD FD RD BU LU FU RU LB RB LF RF FRU BUR BLU FUL FDR FLD BDL BRD

15f 720       30
16f 14508     615
17f 226036    9646
18f 2120898   90394
19f 1618793   69802
20f 357       49

DL BL DR FR UR FL UL BR BD BU FU FD UFR URB UBL ULF DRF DFL DLB DBR

15f 552       24
16f 13722     585
17f 241910    10171
18f 2504080   105523
19f 1221024   51925
20f 24        4

UR DF UL DB DL UB DR UF LF RF LB RB FDR FRU BDL BLU BRD FUL FLD BUR

14f 64        8
15f 1204      151
16f 22362     2836
17f 354750    45087
18f 2610544   331516
19f 992332    127268
20f 56        14

UL UF UR UB DL DF DR DB RF LF RB LB FUL RUF BUR LUB FLD LBD BRD RFD

14f 16        2
15f 512       64
16f 12452     1567
17f 265113    33378
18f 2611642   329002
19f 1091498   138842
20f 79        25

DR RF DL LB UL RB UR LF DF UF UB DB UFR URB UBL ULF DRF DFL DLB DBR

14f 64        8
15f 1040      132
16f 18338     2337
17f 312168    39681
18f 2549550   322317
19f 1100104   140093
20f 48        8

FU RF BU LB FD RB BD LF RD LU RU LD ULF UFR URB UBL DBR DRF DFL DLB

15f 624       80
16f 16050     2050
17f 279938    35657
18f 2406252   305501
19f 1278378   163575
20f 70        17

UB RD UF LD FD LU BD RU LB RB LF RF FRU BUR BLU FUL LBD RDB RFD LDF

15f 496       124
16f 10896     2791
17f 197241    49936
18f 2218817   560150
19f 1553826   395614
20f 36        25

I want to thank Michael Reid who's solver has been of great help.