Parity PLL

Normal PLL (Permutation of the Last Layer) contains only even permutations. However, in certain puzzles (for instance, the Square-1,the 4x4x4 and some cuboids) this restriction is absent because of the existence of parity. This means that on these puzzles we can get a PLL that is not one of the normal 21 found on the 3x3x3 - that is, we can get a Parity PLL.

The following page gives a list of all the parity PLLs, along with some tentative names. Unlike normal PLL, there is no official/traditional naming scheme, probably because few people have bothered to learn algorithms for each of these cases. These cases can simply be solved by doing a parity algorithm and then solving the normal PLL case, but it may be faster to have a one-look alg for them, depending on the puzzle.

All Parity PLL Cases

All Corners Solved Cases

I Permutation Other Names: Pure-E Permutation, Opposite Parity	Adjacent Edges Permutation Other Names: // Permutation, Double Slash Permutation, Adjacent-E Permutation, Adjacent Parity	W Permutation Other Names: B Permutation
Oa Permutation Other Names: O- Permutation, CcwO Permutation	Ob Permutation Other Names: O+ Permutation, CwO Permutation

Adjacent Corner Swap Cases

Adjacent Corners Permutation Other Names: Pj Permutation	Pa Permutation Other Names: Ka Permutation	Pb Permutation Other Names: Kb Permutation
Ka Permutation	Kb Permutation	Ba Permutation Other Names: RGc Permutation
Bb Permutation Other Names: RGa Permutation	Da Permutation Other Names: RGb Permutation	Db Permutation Other Names: RGd Permutation
Ca Permutation	Cb Permutation	M Permutation

Diagonal Corners Swap Cases

Diagonal Corners Permutation Other Names: Slash Permutation, \ Permutation	Q Permutation Other Names: θ Permutation, Theta Permutation	X Permutation Other Names: Ξ Permutation, Xi Permutation
Sb Permutation	Sa Permutation

How Parity Algorithms Affect Cases

In speedsolving, you can either learn specific algorithms for each Parity PLL or you can learn what regular PLL you are gonna get after doing the regular opposite swap parity algorithm to reduce reaction time.

Generally, an opposite swap of edges is the fastest parity algorithms in a puzzle, therefore you should learn based on your preferences on what side you should perform the parity algorithm.

Note that in the following table, the orientation on which side you will get the PLL is correct, but the colors are not.

Parity PLL	PLL after vertical opposite edge swap	PLL after horizontal opposite edge swap
I Permutation	Solves the layer	H Permutation
Adjacent Edges Permutation	Ub Permutation	Ua Permutation
W Permutation	Ub Permutation	Ua Permutation
Oa Permutation	Z Permutation	Z Permutation
Ob Permutation	Z Permutation	Z Permutation
Adjacent Corners Permutation	F Permutation	T Permutation
Pa Permutation	Ja Permutation	Ga Permutation
Pb Permutation	Jb Permutation	Gc Permutation
Ka Permutation	Ab Permutation	Aa Permutation
Kb Permutation	Aa Permutation	Ab Permutation
Ba Permutation	Gc Permutation	Jb Permutation
Bb Permutation	Ga Permutation	Ja Permutation
Da Permutation	Gd Permutation	Ra Permutation
Db Permutation	Gb Permutation	Rb Permutation
Ca Permutation	Rb Permutation	Gb Permutation
Cb Permutation	Ra Permutation	Gd Permutation
M Permutation	T Permutation	F Permutation
Diagonal Corners Permutation	I Permutation	I Permutation
Q Permutation	E Permutation	E Permutation
X Permutation	E Permutation	E Permutation
Sa Permutation	Y Permutation	V Permutation
Sb Permutation	Y Permutation	V Permutation