Bruce MacKenzie
Member
There are 4! * 4! / 2 = 288 PLL permutations which you speed-solvers reduce down to 22 equivalence classes ( 21 + the trivial class containing the identity cube). I wanted to see if I could duplicate this degree of reduction.
I generated the 288 PLL permutations by closing the group formed from the generators ( U , Jb and Aa ). These may be partitioned into 84 symmetry equivalence classes by conjugation with the C4y rotation symmetry group. In speed-solving two PLL permutations which differ only by turns of the U face are deemed to be the same permutation. So for each element of an eq class I applied U, U' and U2 and added those elements to that class. That done the 84 symmetry eq classes collapsed to the expected 22 classes. I then pulled a representative element from each class and compared them to those in the database here. I was gratified that my permutations matched.
My results are given below. The algorithm I used to pull out the representative element gives priority to perms with the UBL cubie solved, then the UL cubie and then the UB cubie. This often gives a differently oriented element than that in the database. The Generator column gives a turn sequence which will take a solved cube to the permutation and the Solution column gives a turn sequence to take the permutation to the solved cube. The Cycle column describes the action of the solution permutation. e.g (UFR, URB, ULF) is read as: the cubie in the URB slot moves to the UFR slot, the cubie in the ULF slot moves to the URB slot and the cubie in the UFR slot moves to the ULF slot. The turn sequences are the output of my optimal solver and make no presumption of being "finger friendly".
I generated the 288 PLL permutations by closing the group formed from the generators ( U , Jb and Aa ). These may be partitioned into 84 symmetry equivalence classes by conjugation with the C4y rotation symmetry group. In speed-solving two PLL permutations which differ only by turns of the U face are deemed to be the same permutation. So for each element of an eq class I applied U, U' and U2 and added those elements to that class. That done the 84 symmetry eq classes collapsed to the expected 22 classes. I then pulled a representative element from each class and compared them to those in the database here. I was gratified that my permutations matched.
My results are given below. The algorithm I used to pull out the representative element gives priority to perms with the UBL cubie solved, then the UL cubie and then the UB cubie. This often gives a differently oriented element than that in the database. The Generator column gives a turn sequence which will take a solved cube to the permutation and the Solution column gives a turn sequence to take the permutation to the solved cube. The Cycle column describes the action of the solution permutation. e.g (UFR, URB, ULF) is read as: the cubie in the URB slot moves to the UFR slot, the cubie in the ULF slot moves to the URB slot and the cubie in the UFR slot moves to the ULF slot. The turn sequences are the output of my optimal solver and make no presumption of being "finger friendly".
| Name | Generator | Solution | Cycle Notation |
| | | | |
1 | A-PLL a | F2 R2 F L F' R2 F L' F | R2 F2 R' B' R F2 R' B R' | (UFR, URB, ULF) |
| | | | |
2 | A-PLL b | R2 F2 R' B' R F2 R' B R' | F2 R2 F L F' R2 F L' F | (UFR, ULF, URB) |
| | | | |
3 | E-PLL | F B R F' L F R' F' B' R B L' B' R' | F B R F' L F R' F' B' R B L' B' R' | (UFR, ULF) (URB, UBL) |
| | | | |
4 | F-PLL | U R L' B2 L' D R' B2 L U' L R2 F2 R2 | U R L' B2 L' D R' B2 L U' L R2 F2 R2 | (UF, UB) (UFR, URB) |
| | | | |
5 | G-PLL a | B' U' B L2 D F' U F U' F D' L2 | L2 D F' U F' U' F D' L2 B' U B | (UF, UL, UR) (UFR, URB, ULF) |
| | | | |
6 | G-PLL b | L2 D F' U F' U' F D' L2 B' U B | B' U' B L2 D F' U F U' F D' L2 | (UF, UR, UL) (UFR, ULF, URB) |
| | | | |
7 | G-PLL c | L U L' B2 D' R U' R' U R' D B2 | B2 D' R U' R U R' D B2 L U' L' | (UF, UR, UB) (UFR, ULF, URB) |
| | | | |
8 | G-PLL d | B2 D' R U' R U R' D B2 L U' L' | L U L' B2 D' R U' R' U R' D B2 | (UF, UB, UR) (UFR, URB, ULF) |
| | | | |
9 | H-PLL | U R2 F2 B2 L2 D' R2 F2 B2 L2 | U R2 F2 B2 L2 D' R2 F2 B2 L2 | (UF, UB) (UR, UL) |
| | | | |
10 | J-PLL a | L2 B' U' B L2 F' D F' D' F2 | L2 B' U' B L2 F' D F' D' F2 | (UF, UR) (UFR, ULF) |
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11 | J-PLL b | B2 L U L' B2 R D' R D R2 | B2 L U L' B2 R D' R D R2 | (UF, UR) (UFR, URB) |
| | | | |
12 | N-PLL a | U R U' R2 B2 D' L F2 L' D B2 R2 U R' | U R U' R2 B2 D' L F2 L' D B2 R2 U R' | (UR, UL) (URB, ULF) |
| | | | |
13 | N-PLL b | U R U' R2 F2 U' R F2 R' U F2 R2 U R' | U R U' R2 F2 U' R F2 R' U F2 R2 U R' | (UF, UB) (URB, ULF) |
| | | | |
14 | R-PLL a | U' R U' B L' B' R' B L B' U' R U2 R' | U' R U' B L' B' R' B L B' U' R U2 R' | (UL, UB) (UFR, ULF) |
| | | | |
15 | R-PLL b | U' R U B' D2 F L' F' D2 B2 U' R' U B' | U' R U B' D2 F L' F' D2 B2 U' R' U B' | (UF, UL) (UFR, URB) |
| | | | |
16 | T-PLL | F2 U F2 U' F2 L2 U' L2 D F2 D' | F2 U F2 U' F2 L2 U' L2 D F2 D' | (UF, UB) (UFR, ULF) |
| | | | |
17 | U-PLL a | R2 U F B' R2 F' B U R2 | R2 U' F B' R2 F' B U' R2 | (UF, UB, UR) |
| | | | |
18 | U-PLL b | R2 U' F B' R2 F' B U' R2 | R2 U F B' R2 F' B U R2 | (UF, UR, UB) |
| | | | |
19 | V-PLL | R U D2 L' U L U2 F2 D R D' F2 D2 R' | R U D2 L' U L U2 F2 D R D' F2 D2 R' | (UF, UR) (URB, ULF) |
| | | | |
20 | Y-PLL | F' R' F L D' F' D R F L' F2 U F2 | F' R' F L D' F' D R F L' F2 U F2 | (UR, UB) (URB, ULF) |
| | | | |
21 | Z-PLL | R2 U R2 U' R2 F2 R2 U' F2 U R2 F2 | R2 U R2 U' R2 F2 R2 U' F2 U R2 F2 | (UF, UL) (UR, UB) |
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