Permuting The Last Layer

The graphics show the top view of the last unsolved layer (the U side). For some positions, there are several algorithms, all doing the same thing. Choose among them according to your preference. The numbers in brackets show the number of moves for each algorithm in four different countings (face moves, quarter moves, slice moves, and antislice moves). More details and notation explanation can be found in the legend or at the bottom of this page. Each permutation is associated with a letter that loosely corresponds to the permutation pattern to allow easy identification of the algorithms. OK, it does require some creativity sometimes, but never the less, I think it is still useful. Think of stellar constellations and their names as a good example. This classification has been proposed by Mirek Goljan back in the early 80's and we find it quite useful when discussing the cube.

Name Permutation Algorithm(s) Probability
of Occurrence
U 
 (9,12,7,4) R²U Fs R²Bs U R²
1/9
A 

 (9,12,9,3) L F'L B²L'F L B²L²
1/9
Z 

 (12,18,7,6)  Ls Ds2 Ls D   Ls2 U' Bs2 
 (12,13,11,3) R' F   R  Ba' R   Fs R  F'R'B2 (U3)
 Ls D L2 F2 R2 L2 B2 R2 D' Ls
1/36
H 

 (10,12,10,5) Ra U²  Ra'Fa'U² Fa 
 (9,17,7,5)   R2 Bs2 L² D' R²Bs2 L²(U)
1/72
E 

 (14,14,14,4) R B L B'R'Fa R F'L'F R'Fa' 
 (15,16,15,4) F R'F'L F R  F'L²B'R B L B'R'B
1/36
T 

 (10,16,10,5) (U3)L²D F²D'L²B²D'R²D B² 
 (14,15,14,3) R B U'B'U B U B²R'B U B U'B'
1/18
V 

 (15,17,15,3) L'U R U'L U L'U R'U'L U²R U²R'
 (14,15,14,5) F'U F'U'R'D R'D'R²F'R'F R F 
 (14,19,14,4) (U2)B U B²R B²U'B'R²F'U F R²U²R'
1/18
F 

 (14,17,14,5) L²F'L D²R'B R D²L B L F L'B'
 (13,18,12,6) (U3)R'L F²L D'R F²L'U R²L'B²R²
 (13,17,12,6) (U2)B L'F U²B'R B'F²D²F²R'B F'
 (16,16,16,4) (U1)F R U'B U F'B'U B U'F R'F'R B'R'
1/18
R 

 (13,14,13,4) (U')B'U²B U'R'F R B'R'F'R U'B
 (13,17,13,5) (U2)R'F²L²D'L D L F²R U²L U'L'
 (14,15,14,3) F L U L'F L U'F U F U'F'L'F²
1/9
J 

 (10,13,10,5)  B2 L  U L'B²R D'R D R²
 (10,12,10,3) (U')Fa U²B'U'B U²F'U B'
1/9
Y 

 (13,15,13,5) R B U'B'R D B'L'B'L B²D'R²
 (14,15,14,4) (U2)L²U L'F L U'L'U'F'U'L'B'U B
 (13,18,13,4) (U3)B L B'R²B L'B'U²R²U'R²U'R²
1/18
G 

 (12,14,12,5) (U2)L U'R U²L'U R'Fa'U²F B
 (12,14,12,5) F U F'L²D'B U'B'U B' D L²
2/9
N 

 (14,17,14,5) L D'B L'D²R F' R'D²L²B'L'D L' 
 (13,18,13,5) R D'F²L²D'L B² L'D L²F²D R'(U1)
 (14,16,13,5) L'U R'U²L U'Rs U R'U²L U'R (U1)
1/36

Legend:

U = Top(Up) D = Down (Bottom)
R = Right   L = Left
B = Back    F = Front

Ls = LR'   Rs = RL'   Ra = RL = La
Ds = DU'   Us = UD'   Da = DU = Ua
Fs = FB'   Bs = BF'   Fa = FB = Ba

Ls' = Rs   Rs' = Ls   Ra' = R'L' = La'
Ds' = Us   Us' = Ds   Da' = D'U' = Ua'
Fs' = Bs   Bs' = Fs   Fa' = F'B' = Ba'

Inverting:  Read backwards and negate everything.

Reflecting: 
  over x axis replace
      B  - F'  F  - B'  R  - R'  L  - L'  D  - D'  U  - U'
      B' - F   F' - B   R' - R   L' - L   D' - D   U' - U
      B2 - F2  F2 - B2  R2 - R2  L2 - L2  D2 - D2  U2 - U2
      Bs - Bs  Fs - Fs  Rs - Ls  Ls - Rs  Ds - Us  Us - Ds
      Ba - Ba' Fa - Fa' Ra - Ra' La - La' Da - Da' Ua - Ua'

  over y axis replace
      R  - L'  L  - R'  F  - F'  B  - B'  D  - D'  U  - U'
      R' - L   L' - R   F' - F   B' - B   D' - D   U' - U
      R2 - L2  L2 - R2  F2 - F2  B2 - B2  D2 - D2  U2 - U2
      Rs - Rs  Ls - Ls  Fs - Bs  Bs - Fs  Ds - Us  Us - Ds
      Ra - Ra' La - La' Fa - Fa' Ba - Ba' Da - Da' Ua - Ua'

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