# Entire Set of Speedsolving Wiki 2-Cycle PLL Algorithms Decomposed

#### Christopher Mowla

(If this thread should be in another forum, mods feel free to move it where it fits best. I wasn't sure.)

I was looking at the PLL Page in the wiki, and I decided to rewrite all 2-cycle cases as commutators and conjugates (along with the extra quarter turn, of course) (one or two algorithms actually is just a conjugate + extra quarter turn!).

If those who edit the wiki wish to put these decompositions next to their corresponding algorithms in the wiki, you are more than welcome too! If not, then I guess a link to this thread would be fine.

The main idea of this post is me giving decompositions for every algorithm for every 2-cycle PLL case currently listed in the wiki (if you want me to decompose a new algorithm, feel free to post) and talking about some way to hopefully organize the algorithms listed for each case better, because...

Before I looked at the wiki, I assumed that there were no repeats, but for most of the cases, there were some algorithms which were speed optimized differently (using different cube rotations and/or wide turn equivalences) and thus several algorithms are repeated multiple times in the list (often times they are not even next to each other in the list).

I will list the decompositions I found, and I will also label how algorithms currently listed in the wiki are related.

I number each algorithm based on their current position in the list (the algorithm in the top of the list has a number [1] next to it, etc.). For cases which have a mirror case (J, N, and R Perms) I have "[1a]" as the number for the first algorithm in the "J Permutation: a" list, for example. Of course the algorithm lists in the wiki will change, and so I identify each algorithm in the following format.

1) Algorithm written exactly how it appears in the wiki (I have added cube rotations at the end of those that needed it so that the cube is restored to its original position).
2) Algorithm without cube rotations, parenthesis grouping, etc. to match the PLL case image exactly. = "unoptimized version."
3) Decomposition of (2).

For algorithms which appear more than once (the algorithms which have more than one optimized form), I have grouped them together in the following manner. The algorithms might not be listed in numerical order, but that is simply because there is currently no order as far as this is concerned in wiki anyway.

1) Version 1 of algorithm exactly how it appears in the wiki
2) Version 2 of algorithm exactly how it appears in the wiki
3) Version 3 of algorithm exactly how it appears in the wiki
...
n) Version n of algorithm exactly how it appears in the wiki
2) Algorithm without cube rotations, parenthesis grouping, etc. to match the PLL case image exactly.
3) Decomposition of (2).

There are three instances where an algorithm has been written twice (exact same form and everything). I have colored the second occurrence in red, so that someone will see it and delete it.

For the cases with mirrors (J, N, and R Perms), anyone could guess that since the number of algorithms is not equal in the a and b versions of these cases, that they don't have the same number of algorithms listed under them, even though they are the same case. I have included the mirror (for J and N perms) or the mirror + y2 cube rotation (R Perm) in blue font, and I recommend that they be added to either the a or b list.

I also labeled algorithms which are inverses of others or those which are just a U or U' conjugate of another algorithm (where the two algorithms which have this relationship either begin or end with U or U': they are a cyclic shift of either U or U' from each other), as well as inverses.

Also, when I say "Mirror," I am referring to mirroring L/R.

In short, I made sure that every unique algorithm (disregarding its inverse, and multiple of occurrence of itself or inverse. In addition, I don't count a U or U' conjugation as a different algorithm) has a mirror for the J, N, and R Permutations so that there will be a copy of every entirely unique algorithm in both lists a and b.

I didn't want to make any changes to the wiki myself because I assume that those who periodically alter the wiki are aware of the multiple occurrences of the same algorithm in different optimized forms, but I don't really see why more than two versions of the same algorithm (and more than two versions of its inverse) are necessary. Oh God I hated the V Permutation list.:fp

I believe I show enough information about the relationships between the algorithms so that people who edit the wiki can make a decision as to what versions of THE SAME EXACT ALGORITHM should be kept, and possibly to make the order of the algorithms more organized. In addition, that the lists of the a and b versions of the mirror cases to directly correspond.

I know that this might sound odd, but I think that if only the unoptimized version of the algorithm is posted on the wiki page, then no one would have an excuse to post a duplicate in the future. AND, to show multiple optimized versions of the algorithm, the links to the algorithms can have all versions execute one after another in alg.garron.us (including the unoptimized version).

Before I show my results, I want to mention that I wrote "(Very Hard)" next to the decompositions of the algorithms for which I had the most trouble finding decompositions for. So for those who want a challenge, try to decompose those algorithms (because I have to say that most of the J Perms and F Perms were a joke).

IMPORTANT NOTES!!!

• For the decompositions, please look at the unoptimized version of the algorithm (the version with no cube rotations) applied on a 3x3x3 Supercube. The center which is rotated 90 degrees (in either direction) is the face for which the extra quarter turn is. So you will know which moves are candidates to be the extra quarter turn, even before trying to decompose the odd permutation PLL.

• In the decompositions, I have put the extra quarter turn in parenthesis. For example, if the right face's center is rotated 90 degrees, then we should know right away that the extra quarter turn is from one of the R moves in the unoptimized version of the algorithm. Note that there can be more than one brief decomposition of an algorithm, especially if the algorithm has more than one occurrence of the face turned in the direction that causes a particular (one) center to be turned 90 (or -90) degrees.
Now, the only decomposition which I believe needs explanation is the following algorithm (and there are several which are directly related to it for which you can follow this example to arrive at the decompositions I have given them).

The first N Permutation a algorithm in the wiki:
(z) D R' U R2 D' R (U' D) R' U R2 D' R U' R' (z')
= R U' L U2 R' U L' R U' L U2 R' U L' U'
=
(R U' L U2 R' U L')2 U'
=
R U'
L U2 R' U L'
R U'
L U2 R' U L'
(U')
=
R L L' U'
L U U' U2 R' U L'
R U' U U'
L U U' U2 R' U R R' L'
(U')
=
R L
L' U' L U
U2
U'
R' U R U'
U
L' U' L U
U2
U' R' U R
L' R'
(U')
= [R L: [L', U'] [U2: [U': [R', U] ] [L', U'] ] [U', R'] ] (U')

[1] R' U R U' R2 (y') R' U' R U (y x) R U R' U' R2 (x') U'
R' U R U' R2 F' U' F U R F R' F' R2 U'
[R', U] [R2: [F', U'] [R, F] ] (U')

[2] (y2) R' U2 R' d' R' F' R2 U' R' U R' F R U' F y
L' U2 L' U' F' L' F2 U' F' U F' L F U' L
[L' U: [U: (L')] [F' L' F: [F, U'] ] ]

[3] (y') F r2 R' U2 r U' r' U2 x' R2 U' R' U r2 u' z' y2
L B2 F' D2 B R' B' D2 F2 L' F' L B2 L'
[L B2: [F' D2 B: (R')] [F, L'] ]

[4] (y) R' U' F' R U R' U' R' F R2 U' R' U' R U R' U R y'
B' U' R' B U B' U' B' R B2 U' B' U' B U B' U B
[B' U': [R': [B, U] [B', R] ] [B U' B': (U')] ]

[5] (y2) R U' R' U R2 y R U R' U' x U' R' U R U2 (x') U y
L U' L' U L2 F U F' U' L' F' L F L2 U
[L, U'] [L2: [F, U] [L', F'] ] (U)
::::::::::::::::::::::::
::::::::A U conjugation of [5]
[6] (y' z) R U R' U' R U2 (z' y') R U R' U' (y x) L' U' L U L2 x' y
U B U' B' U B2 L U L' U' B' L' B L B2
(U) [B, U'] [B2: [L, U] [B', L'] ]

[7] R' U2 R' U2 L U' R' U' R' U R U2 L' U2 R U R
R' U2 R' U2 L U' R' U' R' U R U2 L' U2 R U R
[R' U': [R' U2 L U': [U, L' U2 R] [U' R' U': (R')] ] ]

[8] L U2 L U2 R' U L U L U' L' U2 R U2 L' U' L'
L U2 L U2 R' U L U L U' L' U2 R U2 L' U' L'
[L2: [L', U2] [R': [U L U: (L)] [U', R] ] [U, L'] ]

[9] (y2) M U2 r' U l' U2 r U' R2 r U2 R2 (x') U y2
L R' F2 R' D L' F2 R U' L2 R B2 L2 U
[L R': [F2 R': (D)] [F2, L' R'] ] [R', U'] [U' L2: B2] (VERY HARD)

[10] (r U R' U') z U' (l' U2 r' U') (r U2 l' U l2) (x') y
L F R' F' L' F' D2 B' L' B D2 F' R F2
[F2 R': [R, F2 L F] [F D2 B': (L')] ]

[11] (y) F (x) R2 F R2 U' R' U l' U2 L F' L' U (x') R2 U' (x') y'
R B2 D B2 R' B' R B' D2 F L' F' D B2 R'
[R B2 D: [B2: [R', B'] ] [D2 F: (L')] ]

[12] (y) (R U R' U R U2 R2) (U' R U' R' U2 R) U (r U R' U' r' F R F') (y')
B U B' U B U2 B2 U' B U' B' U2 B U F R B' R' F' R B R'
[B: (U)] [U': [U2, B] [B', U'] [U2, B'] ] [F, R B' R']

[13] (y) M2 U M2 (y') (R U R' U' R' F R2 U' R' U' R U R' F') (y) M2 U' M2 y'
F2 B2 D F2 B2 R U R' U' R' F R2 U' R' U' R U R' F B2 D' F2 B2
[F2 B2 D B2 F': [F': [R, U] [R', F] ] [R U' R': (U')] ]

[14] (y) F U' r U r' U r' F' r2 U' r' U' F' U2 F' y'
R U' F R F' U F' U' F2 R' F' U' R' U2 R'
[R U': [F R F': [U, F'] ] [U': (R')] ]

a
b

[1a] B' U F' U2 B U' B' U2 F B U'
[16a] (y') L' U R' U2 L U' L' U2 R L U' y
B' U F' U2 B U' B' U2 F B U'
[B' U: [F' U2 B: (U')] [B, U'] ]
::::::::Mirror of [1a] and [16a]
[4b](y') L U' R U2 L' U L U2 (R' L') U y
[11b](y) R U' L U2 R' U R U2 L' R' U y'
B U' F U2 B' U B U2 F' B' U
[U' B: [B', U] [F U2 B': (U)] ]

[2a] z (y) R2 u' R' u R2 x' y' R' U R' U' R2 z2
[4a] (y2 x) U2 r' U' r U2 l' U R' U' l2 y2
B2 R' U' R B2 L' D L' D' L2
[B2 R': (U')] [L': [D, L'] ]
::::::::Mirror of [2a] and [4a]
[9b](y2) F2 R U R' F2 L D' L D L2 y2
[10b](y2) F2 R U R' b2 R U' R U l2'
B2 L U L' B2 R D' R D R2
[B2 L: (U)] [R: [D', R] ]
::::::::Inverse of [9b] and [10b]
[8b](y2) r2 U' L' U r' U2 R B' R' U2 (x') y2
R2 D' R' D R' B2 L U' L' B2
[R: [R, D'] ] [B2 L: (U')]

[10a] L' U R' U2 L U' R U L' U R' U2 L U' R
L' U R' U2 L U' R U L' U R' U2 L U' R
[L' R': [R, U] [U' L U', R] (U) [U, L] ]
::::::::Mirror of [10a]
[3b] R U' L U2 R' U L' U' R U' L U2 R' U L'
R U' L U2 R' U L' U' R U' L U2 R' U L'
[L R: [L', U'] [U R' U, L'] (U') [U', R'] ]

[13a] L' U' L F L' U' L U L F' L2 U L U
L' U' L F L' U' L U L F' L2 U L U
[L' U' L F: [L', U'] [L, F'] ] (U)
::::::::Mirror
[1b] R U R' F' R U R' U' R' F R2 U' R' U'
R U R' F' R U R' U' R' F R2 U' R' U'
[R U R' F': [R, U] [R', F]] (U')

[14a] B2' R2 U' R2' u R2 D' R2' U R2 U'
B2 R2 U' R2' D B2 D' B2 U B2 U'
[B2: [R2: (U')] [D, B2] [U, B2] ]
::::::::Mirror of [14a] (Should be added to J Perm (b) list)
B2 L2 U L2 u' L2 D L2 U' L2 U
B2 L2 U L2 u' L2 D L2 U' L2 U
[B2: [L2: (U)] [D', B2] [U', B2] ]

[15a] (y) R2 (U' D) R2' U' R2 U R2' D' R2 U R2' U' y'
B2 U' D B2 U' B2 U B2 D' B2 U B2 U'
[D B2: [B2, D'] (U') [B2, U'] ] [B2, U]
::::::::Mirror of [15a] (Should be added to J Perm (b) list)
(y') L2 (U D') L2 U L2 U' L2 D L2 U' L2 U y
B2 U D' B2 U B2 U' B2 D B2 U' B2 U
[D' B2: [B2, D] (U) [B2, U] ] [B2, U']

[3a] (y2) R' U2 R U R' U2' L U' R U L' y2
[6a] (y2) R' U2 R U R' (z) R2' U R' D R U' (z') y2
[11a] L' U2 L U L' U2 R U' L U R'
[18a] y2 R' U2' R U R' z R2 U R' D R U' z' y2
L' U2 L U L' U2 R U' L U R'
[L': [U2 L: (U)] [R U', L] ]
::::::::Inverse of [3a], [6a], [11a], and [18a]
[7a] (y2) L U' R' U L' U2 R U' R' U2 R y2
[9a] R U' L' U R' U2 L U' L' U2' L
[17a] (z) D R' U' R D' R (R U R' U') R2 U z'
R U' L' U R' U U L U' L' U2 L
[L': [R: [L, U'] ] [U2 L: (U')] ]
::::::::Mirror of [3a],[6a],[11a],and [18a]
[2b] R U2 R' U' R U2 L' U R' U' L
R U2 R' U' R U2 L' U R' U' L
[R: [U2 R': (U')] [L': [U, R'] ] ]
::::::::Inverse of [2b]
[5b] L' U R U' L U2 R' U R U2' R'
L' U R U' L U2 R' U R U2' R'
[R: [L': [R', U] ] [U2 R': (U)] ]

[8a] (y2) L' R' U2 R U R' U2 L U' R U y2
[12a] R' L' d2 R U R' d2 R U' L U
R' L' U2 L U L' U2 R U' L U
[L': [R' U2 L: (U)] [U', L] ]
::::::::Inverse of [8a] and [12a]
[5a] U' L' U R' (z) R2 U R' U' R2 U D (z')
U' L' U R' U2 L U' L' U2 L R
[L' R': [R: [L, U'] ] [U2 L: (U')] ]
::::::::Mirror of [8a] and [12a]
[6b](y2) R L U2 L' U' L U2 R' U L' U' y2
L R U2 R' U' R U2 L' U R' U'
[R: [L U2 R': (U')] [U, R'] ]

[19a] (y2) F U' R' F R2 U' R' U' R U R' F' R U R' F' y2
B U' L' B L2 U' L' U' L U L' B' L U L' B'
[B L U' L' B: [B' L U, L'] [L, U'] (U') [U', L] ]
::::::::Mirror of [19a] (Should be added to J Perm (b) list)
(y2) F' U L F' L2 U L U L' U' L F L' U' L F y2
B' U R B' R2 U R U R' U' R B R' U' R B
[B' R' U R B': [B R' U', R] [R', U] (U) [U, R'] ]

[7b] (y') L U' r U2 l' U R' U' l2 F2 L2 y
B U' B L2 F' D F' D' F2 L2 B2
[B2: [B': (U')] [L2 F': [D, F'] ] ]
::::::::Mirror of [7b] (Should be added to J Perm (a) list)
y R' U l' U2 r U' L U r2 F2 R2 y'
B' U B' R2 F D' F D F2 R2 B2
[B2: [B: (U)] [R2 F: [D', F] ] ]

a
b

[2a] (z) U R' D R2 U' R (D' U) R' D R2 U' R D' R' (z')
[10a] (L U' R U2 L' U R')2 U'
(L U' R U2 L' U R')2 U'
[L R: [R', U'] [U2: [U': [L', U] ] [R', U'] ] [U', L'] ] (U') (VERY HARD)
::::::::Mirror of [2a] and [10a]
[4b] (z) (D' R U' R2' D R' U)2 R (z')
[7b] (R' U L' U2 R U' L)2 U
(R' U L' U2 R U' L)2 U
[L' R': [L, U] [U2: [U: [R, U'] ] [L, U] ] [U, R] ] (U) (VERY HARD)
:::::::A U' conjugation of [2a] and [10a]
[9a](y z) R' U R' D R2 U' R (U D') R' D R2 U' R D' (z') y'
U' (F U' B U2 F' U B')2
(U') [F B: [B', U'] [U2: [U': [F', U] ] [B', U'] ] [U', F'] ] (VERY HARD)
::::::::::::::::::::::::
::::::::A U conjugation of [4b] and [7b]
[13b] U (B' U F' U2 B U' F)2
U (B' U F' U2 B U' F)2
(U) [F' B': [F, U] [U2: [U: [B, U'] ] [F, U] ] [U, B] ] (VERY HARD)

[1a] (z) D R' U R2 D' R (U' D) R' U R2 D' R U' R' (z')
[4a] R U' L d2 L' U L R' U' R U2 r' F l' U' (y2)
(R U' L U2 R' U L')2 U'
[R L: [L', U'] [U2: [U': [R', U] ] [L', U'] ] [U', R'] ] (U') (VERY HARD)
::::::::Mirror of [1a] and [4a]
[1b] (z) U' R D' R2' U R' (U' D) R D' R2 U R' D R (z')
[3b] (z) U' R D' R2 U R' U' (z') R U R' (z) R2 U R' (z') R U
[8b] L' U R' U2' L U' (R L') U R' U2' L U' R U
(L' U R' U2' L U' R)2 U
[L' R': [R, U] [U2: [U: [L, U'] ] [R, U] ] [U, L] ] (U) (VERY HARD)

[8a] (z) U R' U' R U F R F' U' R' U F' U F U' R U' (z')
[12a] L U' L' U L F U F' L' U' L F' L F L' U L'
L U' L' U L F U F' L' U' L F' L F L' U L'
[L U': [L' U L F: (U)] [F', L] ]
::::::::Mirror of [8a] and [12a]
[5b] R' U R U' R' F' U' F R U R' F R' F' R U' R
R' U R U' R' F' U' F R U R' F R' F' R U' R
[R' U: [R U' R' F': (U')] [F, R'] ]
::::::::Inverse of [5b]
[12b] (R' U R' F) R F' (R U' R') F' U (F R U R' U') R
R' U R' F R F' R U' R' F' U F R U R' U' R
[R' U: [R', F] [R U' R' F': (U)] ]

[11a] R U' R' U l U F U' R' F' R U' R U l' U R'
R U' R' U R B U B' R' U' R B' R B R' U R'
[R U': [R' U R B: (U)] [B', R] ]
::::::::Mirror of [11a]
[9b] L' U L U' r' U' F' U L F L' U L' U' r U' L
L' U L U' L' B' U' B L U L' B L' B' L U' L
[L' U: [L U' L' B': (U')] [B, L'] ]

[3a] R' U R2 B2 U R' B2' R U' B2 R2' U' R U'
R' U R2 B2 U R' B2' R U' B2 R2' U' R U'
[R' U R2 B2 U R': B2] (U')
::::::::Mirror of [3a] (Should be added to N Permutation (b) list)
L U' L2 B2 U' L B2 L' U B2 L2 U L' U
L U' L2 B2 U' L B2 L' U B2 L2 U L' U
[L U' L2 B2 U' L: B2] (U)

[5a] F' R U R' U' R' F R2 F U' R' U' R U F' R'
F' R U R' U' R' F R2 F U' R' U' R U F' R'
[F' R U, R'] [R F U' R': (U')]
::::::::Mirror of [5a] (Should be added to N Permutation (b) list)
F L' U' L U L F' L2 F' U L U L' U' F L
F L' U' L U L F' L2 F' U L U L' U' F L
[F L' U', L] [L' F' U L: (U)]

[6a] R U R' U (R U R' F' R U R' U' R' F R2 U' R') U2 R U' R'
R U R' U R U R' F' R U R' U' R' F R2 U' R' U2 R U' R'
[R U R': [U2: [U', R] [F' R U, R'] [R, U'] ] (U')]
::::::::Mirror of [6a] (Should be added to N Permutation (b) list)
L' U' L U' (L' U' L F L' U' L U L F' L2 U L) U2 L' U L
L' U' L U' L' U' L F L' U' L U L F' L2 U L U2 L' U L
[L' U' L: [U2: [U, L'] [F L' U', L] [L', U] ] (U)]

[7a] L U L' (y') (R U R' F' R U R' U' R' F R2 U' R') (y') U' R U' R' y2
R U R' B U B' R' B U B' U' B' R B2 U' B' U' R U' R'
[R U R': [B U B' R': [B, U] [B', R]] (U')]
::::::::Mirror of [7a] (Should be added to N Permutation (b) list)
R' U' R (y) (L' U' L F L' U' L U L F' L2 U L) (y) U L' U L y2
L' U' L B' U' B L B' U' B U B L' B2 U B U L' U L
[L' U' L: [B' U' B L: [B', U'] [B, L']] (U)]

[13a] (z) U2 R2 U' R2 (y') R2 U2 R2 U R2 (y) R2 U2 R2 d R2 (x' y)
L2 U2 L' U2 F2 L2 F2 L F2 U2 L2 U2 L F2
[L: [L, U2] [F2, L2] ] [F2: [U2, L2] (L')]
::::::::Mirror of [13a] (Should be added to N Permutation (b) list)
(z') U2 L2 U L2 (y) L2 U2 L2 U' L2 (y') L2 U2 L2 d' L2 (x' y')
R2 U2 R U2 F2 R2 F2 R' F2 U2 R2 U2 R' F2
[R': [R', U2] [F2, R2] ] [F2: [U2, R2] (R)]

[2b] (L' U' L U) R' U2' R U R' z R2 U R' D R U' (R' U' R U) z'
L' U' L U R' U2 R U R' U2 L U' R U L' U' L' U L
[L' U' L U: [R': [U2 R: (U)] [L U', R] ] ]
::::::::Mirror of [2b] (Should be added to N Permutation (a) list)
(R U R' U') L U2 L' U' L z' L2 U' L D' L' U (L U L' U') z
R U R' U' L U2 L' U' L U2 R' U L' U' R U R U' R'
[R U R' U': [L: [U2 L': (U')] [R' U, L'] ] ]

[6b] 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' U
[R U' R2 F2 U' R: F2] (U)
::::::::Mirror of [6b] (Should be added to N Permutation (a) list)
L' U L2 F2 U L' F2 L U' F2 L2 U' L U'
L' U L2 F2 U L' F2 L U' F2 L2 U' L U'
[L' U L2 F2 U L': F2] (U')

[10b] R2 F2 U2 F2 U F2 R2 U2 R2 U F2 U2 R2 U
R2 F2 U2 F2 U F2 R2 U2 R2 U F2 U2 R2 U
[R2 U' F2: [F2, U] [U', F2] [R2, U2] [U', F2] ] (U)
::::::::Mirror of [10b] (Should be added to N Permutation (a) list)
L2 F2 U2 F2 U' F2 L2 U2 L2 U' F2 U2 L2 U'
L2 F2 U2 F2 U' F2 L2 U2 L2 U' F2 U2 L2 U'
[L2 U F2: [F2, U'] [U, F2] [L2, U2] [U, F2] ] (U')

[11b] (R' U' R U') (L U' R' U) L' U2 (R U' R' U2) R (U R' U R)
R' U' R U' L U' R' U L' U2 R U' R' U2 R U R' U R
[R' U' R U': [R': [L: [R, U'] ] [U2 R: (U')] ] ]
::::::::Mirror of [11b] (Should be added to N Permutation (a) list)
(L U L' U) (R' U L U') R U2 (L' U L U2) L' (U' L U' L')
L U L' U R' U L U' R U2 L' U L U2 L' U' L U' L'
[L U L' U: [L: [R': [L', U] ] [U2 L': (U)] ] ]

a
b

[2a] R U2 R' U2 R B' R' U' R U l U R2 F (x)
[5a] (y2 z) U R2 U' R2 U F' U' R' U R U F U2 R (z') y2
[9a] (y2) L U2 L' U2 L F' L' U' L U L F L2 U y2
R U2 R' U2 R B' R' U' R U R B R2 U
[R, U2] [R B' R', U' R U] [U' R: (U)]
::::::::Inverse of [2a], [5a], and [9a]
[12a] (y2) U' L2 F' L' U' L' U L F L' U2 L U2 L' y2
U' R2 B' R' U' R' U R B R' U2 R U2 R'
[U' R: (U')][U' R U, R B' R'] [U2, R]
::::::::Mirror (and y2 cube rotation) of [2a], [5a] and [9a].
[1b] R' U2 R U2 R' F R U R' U' R' F' R2' U'
R' U2 R U2 R' F R U R' U' R' F' R2' U'
[R', U2] [R' F R, U R' U'] [U R': (U')]
::::::::Inverse of [1b]
[8b] U R2 F R U R U' R' F' R U2 R' U2 R
U R2 F R U R U' R' F' R U2 R' U2 R
(U) [R': [R', F R U] ] [U2, R']
::::::::::::::::::::::::
::::::::a U' Conjugation of [8b]
[6b] (y x) R2 U l U R U' l' U' l U2 R' U2 R U
[7b] (y') r' L' U r U L U' r' U' r U2 L' U2 L U y
B2 R B U B U' B' R' B U2 B' U2 B U
[B': [B', R B U] ] [U2, B'] (U)
::::::::Mirror (and y2 cube rotation) of [6b] and [7b]
[6a] (y') R2 B' R' U' R' U R B R' U2 R U2 R' U' y
[11a] (y') R l U' l' U' R' U l U l' U2 R U2' R' U' y
F2 R' F' U' F' U F R F' U2 F U2 F' U'
[F: [F, R' F' U'] [F', U2] ] (U')

[3a] R U' R F2 U R U R U' R' U' F2 R2 U
R U' R F2 U R U R U' R' U' F2 R2 U
[R U': [R F2 U R U: (R)] [R', U] ]
::::::::::::::::::::::::
::::::::Mirror, y2 cube rotation, inverse, and U conjugation of [3a]
[2b] (y) R2 B2 U' R' U' R U R U B2 R U' R U y'
B2 L2 U' B' U' B U B U L2 B U' B U
[U' B' U: [U', B] [B' L2 U' B' U': (B)] ]

[1a] (y') R U R' F' R U2 R' U2 R' F R U R U2 R' U' y
F U F' L' F U2 F' U2 F' L F U F U2 F' U'
[F U F' L' F U2, F'] [U2, F] (U')
::::::::Mirror (and y2 cube rotation) of [1a] (Should be added to R Perm (b) list)
(y) R' U' R B R' U2 R U2 R B' R' U' R' U2 R U y'
B' U' B L B' U2 B U2 B L' B' U' B' U2 B U
[B' U' B L B' U2, B] [U2, B'] (U)

[4a] R U2 R' U' R' F' R U2 R U2 R' F R U' R' U
R U2 R' U' R' F' R U2 R U2 R' F R U' R' U
[R: [U2 R' U2, U R' F' R] ] (U)
::::::::Mirror (and y2 cube rotation) of [4a] (Should be added to R Perm (b) list)
R' U2 R U R B R' U2 R' U2 R B' R' U R U'
R' U2 R U R B R' U2 R' U2 R B' R' U R U'
[R': [U2 R U2, U' R B R'] ] (U')

[7a] (y2) R' U' R U2 R U2 L' R2 U R U' R L U2 R' U2 y2
L' U' L U2 L U2 R' L2 U L U' L R U2 L' U2
[L': (U')] [U2, L] [R': [L', U] ] [L, U2]
::::::::Mirror (and y2 cube rotation) of [7a] (Should be added to R Perm (b) list)
(y2) R U R' U2 R' U2 L R2 U' R' U R' L' U2 R U2 y2
L U L' U2 L' U2 R L2 U' L' U L' R' U2 L U2
[L: (U)] [U2, L'] [R: [L, U'] ] [L', U2]

[8a] R U2 R D R' U R D' R' U' R' U R U R' U
R U2 R D R' U R D' R' U' R' U R U R' U
[U' R: [R', U] [U', R D R'] [U' R': (U)] ]
::::::::Mirror (and y2 cube rotation) of [8a] (Should be added to R Perm (b) list)
R' U2 R' D' R U' R' D R U R U' R' U' R U'
R' U2 R' D' R U' R' D R U R U' R' U' R U'
[U R': [R, U'] [U, R' D' R] [U R: (U')] ]

[10a] (y') R U' R' U' R U R D R' U' R D' R' U2 R' U' y
F U' F' U' F U F D F' U' F D' F' U2 F' U'
[F: [U' F': (U')] [F D F', U'] [U, F'] ]
::::::::Mirror (and y2 cube rotation) of [10a] (Should be added to R Perm (b) list)
(y) R' U R U R' U' R' D' R U R' D R U2 R U y'
B' U B U B' U' B' D' B U B' D B U2 B U
[B': [U B: (U)] [B' D' B, U] [U', B] ]

[13a] (y x') R' U' F' U R' U' (x) U R' U' R' U R B R2 y'
B' L' U' L B' L' U B' U' B' U B L B2
[B2 L': [L B L', U'] [B' U': (B')] ]
::::::::Mirror (and y2 cube rotation) of [13a] (Should be added to R Perm (b) list)
(y' x) R U B U' R U (x') U' R U R U' R' F' R2 y
F L U L' F L U' F U F U' F' L' F2
[F2 L: [L' F' L, U] [F U: (F)] ]

[3b] R' U2 R U' (y') R' F R B' R' F' R (z x') R' U R' (x z') y
R' U2 R U' F' L F R' F' L' F U' R U'
[R' U R: [R', U] [F' L F, R'] ] (U')
::::::::Mirror (and y2 cube rotation) of [3b] (Should be added to R Perm (a) list)
R U2 R' U (y) R B' R' F R B R' (z x) R U' R (x' z') y'
R U2 R' U B L' B' R B L B' U R' U
[R U' R': [R, U'] [B L' B', R] ] (U)

[4b] (y2) R' U2 l R U' R' U l' U2' R F R U' R' U' R U R' F' y2
L' U2 L2 F' L' F L' U2 L B L U' L' U' L U L' B'
[L' U2 L: [L, F'] ] [B L: [U' L': (U')] ]
::::::::Mirror (and y2 cube rotation) of [4b] (Should be added to R Perm (a) list)
(y2) R U2 l' R' U R U' l U2 R' B' R' U R U R' U' R B y2
L U2 L2 B L B' L U2 L' F' L' U L U L' U' L F
[L U2 L': [L', B] ] [F' L': [U L: (U)] ]

[5b] R' U2 R' D' R U' R' D R U R U' R' U' R U'
R' U2 R' D' R U' R' D R U R U' R' U' R U'
[R' U2: [R' D' R, U'] [R, U'] ] (U')
::::::::Mirror (and y2 cube rotation) of [5b] (Should be added to R Perm (a) list)
R U2 R D R' U R D' R' U' R' U R U R' U
R U2 R D R' U R D' R' U' R' U R U R' U
[R U2: [R D R', U] [R', U] ] (U)

[1] R U R' U' R' F R2 U' R' U' R U R' F'
R U R' U' R' F R2 U' R' U' R U R' F'
[R, U] [F: [F', R'] [R U' R': (U')] ]
::::::::Inverse of [1]
[3] F R U' R' U R U R2 F' R U R U' R'
[5] (y2) B L U' L' U L U L2 B' L U L U' L' y2
F R U' R' U R U R2 F' R U R U' R'
[F: [R U' R': (U)] ] [F, R'] [U, R]

[2] (y2) L' U' L U L F' L2 U L U L' U' L F y2
R' U' R U R B' R2 U R U R' U' R B
[R', U'] [B': [B, R] [R' U R: (U)] ]

[4] R2 U R2 U' R2' U' D R2 U' R2' U R2 D'
[9] R2 U R2' U' R2 U' D R2' U' R2 U R2' D'
R2 U R2' U' R2 U' D R2' U' R2 U R2' D'
[R2, U] [D R2: [R2, D'] [U' R2: (U')] ]

[6] (y2) L2 U' L2 D F2 R2 U R2 D' F2 U y2
[7] R2 U' R2 D B2 L2 U L2 D' B2 U
R2 U' R2 D B2 L2 U L2 D' B2 U
[U': [U, R2] [B2 D: [D', B2] [L2: (U)] ] ]

[8] R2' u' R2 U R2' (y) R2 u R2' U' R2 U y'
R2 D' F2 U F2 R2 D B2 U' B2 U
[R2 D' F2: (U)] [R2, D'] [B2, U']

[10] (y2 z) U2 r' U2 r U2 (x) U2 r U2 r' U2 R (z') y
R2 D' F2 D R2 B2 D L2 D' B2 U
[R2 D': F2] [B2 D: L2] (U)

[11] R U R' U' R2 D R' U' R' U' R U (z') U2 R' U (z)
R U R' U' R2 D R' U' R' U' R U R2 D' R
[R, U] [R' D R': [R, D'] [R' U' R': (U')] ]

[12] R2 U R2 U' R2 F2 U' F2 D R2 D'
R2 U R2 U' R2 F2 U' F2 D R2 D'
[R2: [U, R2] [F2: (U')] [D, R2] ]

[1] R' U R' d' R' F' R2 U' R' U R' F R F y'
[2] R' U R' (y) U' R' F' R2 U' R' U R' F R F y'
[5] R' U R' U' (y x) R' U' R2 B' R' B R' U R U (x') y'
[6] R' U R' U' (y x) R' U' R2 (x') U' R' U R' F R F y'
[10] R' U R' U' (y) R' F' R2 U' R' U R' F R F y'
[22] R' U R' U' (y) R' F' R2 U' R' U R' F R F (y')
R' U R' U' B' R' B2 U' B' U B' R B R
[R2: [R, U] [B': (R')] [B, U'] [B', R] ]

[3] R' U R' U' (y x) R' F R' F' R2 U' R' U R U (x') y'
[4] R' U R' U' (y x) R' F R' F' R2 U' R' U R U (x') y'
[7] R' U R' U' B' D B' D' B2 R' B' R B R
[8] R' U R' U' (x2 y') R' U R' U' l R U' R' U R U (x') y'
[9] R' U R' U' (x' y) U' R U' R' U2 (y') R' U' R U R (x)
[11] R' U R' U' (y) R' D R' D' R2 F' R' F R F y'
R' U R' U' B' D B' D' B2 R' B' R B R
[R2: [R, U] [B': [D, B'] [B: (R')] ] [B', R] ]

[12] R' U2 R U2 L U' R' U L' U L U' R U L'
[13] R' U2 R U2 L U' R' U r' F r U' R U L'
R' U2 R U2 L U' R' U L' U L U' R U L'
[R', U2] [L U' R' U L': (U)]

[14] R2 U' (B2 U B2) (R D') (R D) R' U R U' R
R2 U' B2 U B2 R D' R D R' U R U' R
[R2: [U', B2] [R D': (R)] [U, R] ]

[15] (y') R' U L U' R U R' U L' U' R U2 L U2 L' y (*)
[18] (y) r' F R F' r U r' F R' F' r U2 R U2 R' y'
[19] (y) L' U R U' L U L' U R' U' L U2 R U2 R' y'
[21] (y z) U' R D R' U R U' R D' R' U R2 D R2 D' z' y'
F' U B U' F U F' U B' U' F U2 B U2 B'
[F' U B U' F: (U)] [U2, B]

[16] (y2) R U' L' U R' U' R U' L U R' U2 L' U2 L y2
[17] L U' R' U L' U' L U' R U L' U2 R' U2 R
L U' R' U L' U' L U' R U L' U2 R' U2 R
[L U' R' U L': (U')] [U2, R']

[20] R U2 R' D R U' R U' R U R2 D R' U' R D2
R U2 R' D R U' R U' R U R2 D R' U' R D2
[U: [U': (R)] [D: [D', U R'] [R: [U', R] ] [D: [R', U']] ] ] (VERY HARD)

[1] F R U' R' U' R U R' F' R U R' U' R' F R F'
F R U' R' U' R U R' F' R U R' U' R' F R F'
[F R U' R': (U')] [R, U] [R', F]

[2] R' U' R F2 R' U R d R2 U' R2' U' R2 y
R' U' R F2 R' U R U F2 U' F2 U' F2
[F2: [F2, R' U' R] [U F2: (U')] ]

[3] R2 U' R2 U' R2 d R U R' B2 R U' R' y
[12] (y2 z) U2 R' U2 R' U2 R B R B' U2 B R' B' (z') y2
R2 U' R2 U' R2 U F U F' R2 F U' F'
[R2: [U' R2: (U')] [F U F', R2] ]

[4] (y') R2 u R2' U R2 D' R' U' R F2' R' U R
F2 D R2 U R2 D' R' U' R F2 R' U R
[F2: [D R2: (U)] [R' U' R, F2] ]

[5] F R' F R2 U' R' U' R U R' F' R U R' U' F'
[16] F R' F R2 U' R' U' R U R' F' R U R' U' F'
F R' F R2 U' R' U' R U R' F' R U R' U' F'
[F2: [F', R'] [R U' R': (U')] [F': [R, U] ] ]

[6] F R' F' R U R U' R2 U' R U l U' R' U F (x)
F R' F' R U R U' R2 U' R U R B' R' B U
[F, R'] [U, R] [R', U'] [R, B'] (U)

[7] F R U' R' U' R d R U R' B' R U' R2 y
F R U' R' U' R U F U F' R' F U' F2
[F: [R, U'] [U2, R] [R: [U', F] ] [F: (U')] ]

[8] R' F R F' (y') U' R' U R2 U R' U' R' F R F' U' y
R' F R F' U' F' U F2 U F' U' F' L F L' U'
[R', F] [U', F'] [F, U] [F', L] (U')

[9] (y) F' L' U L U L' U' L F L' U' L U L F' L' F y'
R' F' U F U F' U' F R F' U' F U F R' F' R
[R' F' U F: (U)] [F', U'] [F, R']

[10] (y z) U2 R U R' U' R (y) R U L' U L U R' U' (x) y2
F2 U F U' F' U L F R' F R F L' F'
[F2: [U, F] (U) [L F: [R', F] ] [L, F'] ]

[11] (z') U2 L' U' L U L' (y') L' U' R U' R' U' L U (x) y
[13] R2 U' R' U R U' (z' y') L' U' R U' R' U' L U (y z)
R2 U' R' U R U' B' R' F R' F' R' B R
[R2: [U, F] (U) [L F: [R', F] ] [L, F'] ]

[14] F R U' R' F D R' (y x) R' U' R (z) R2 (y) L' U2 y2 (z) y
F R U' R' F D R' B' R' B R2 D' F2
[F R: (U')] [F2 D R2: [R, B'] ]

[15] (y2 z) U' R' U2 R' D R2 U' R U R2 U' D' R U' R U (z')
R' U' R2 U' L U2 R' U R U2 L' R' U R' U R
[R' U' R2: [U' L U2 R': (U)] [U', R'] ]

[17] y r U r' U' r U' r' y' l U' R2 U l' U' R2 U
F R F' U' F R' F' R B' R2 B R' U' R2 U
[F R F': (U')] [R: [B', R2] ] [R2, U']

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