CPLS


CPLS (short for Corner Permutation and Last Slot) is a substep used to solve the final F2L corner (usually DFR) and permute the lastlayer corners while preserving the corresponding F2L edge and EOLL. If used when only the last F2L corner, EPLL, CPLL, and COLL remain, CPLS leaves just EPLL and COLL, which may be solved in 2generator. Although seemingly hard to recognize, CPLS can be very beneficial, especially for onehanded solvers if fast 2generator algorithms are used.
A twostep solution to EPLL and COLL requires only 11 algorithms: 7 for OCLL, which leaves 4 EPLL cases. Onestep solution, known as 2GLL, has 84 cases. Note that this is also referred to as ZZd.
History
CPLS was proposed by Baian Liu in 2009.
Learning Approach
CPLS is naturally divided into the subsets  +, O, I, Im, and C, the same classification used for CLS. The first three sets have 6 algorithms each, the next two have 3 each, and the final set has only 2. One recommended order, for ease of learning and recognition, is C, O, I, Im, , +.
Recognition
Although the seemingly difficult recognition may appear to be a possible disadvantage of CPLS, the following system due to Stachu Korick works well.
 AUF the last F2L corner to URF, and note the group (O etc).
 For each of the three Ulayer corners, locate the sticker clockwise from its Ulayer sticker. This is best done in a fixed order (say UFL, UBL, UBR).
 Pretend that the face containing the each sticker is of the same color. The required corner permutation is the permutation of the corresponding faces necessary to regain the standard color scheme.
Scramble: x R' U R U' R' U R U' R' U R U' x' (yellow on top, orange in front)
The last F2L corner is already at URF. This is an O case. The stickers clockwise from Ulayer (yellow) stickers are at FLU (red), LUB (green), BRU (orange). Pretending that F = red, L = green, and B = orange, the required permutation swaps F and B, represented by FLU and BRU corners. Thus we want O with diagonal swap (both diagonal swaps have the same effect), which is x (U R' U' R)*3 x'. We may write this case as [O BLF], meaning the true color of the B face is on F, that of L on L, and F on B.
Example Solves
This recognition method is best illustrated with example solves. Standard color scheme is assumed throughout. Scramble with white on top (U) and green in front (F). Notes in square brackets by Stachu Korick.
Scramble 1: D' B' D2 F2 R F' L' R B L' F' U' R' B F2 U R2 D L2 D2 U' B R' F2 D'
A Petrus approach
2x2x2: x2 D B' R' D2 L2 (5/5) 2x2x3: x' y D R D' U' L' U L (7/12) [Tthis sets up for an Im case later on] EO: y U M' U M (4/16) F2L: x y U2 R' U' R U R' U' R2 U2 R' (10/26) [ewww] CPLS (Im FBL): y2 U R' F2 R D' L' U2 L' U' L2 D (11/37) 2GLL (Pi Ua1): y (y) R U2 R2 U2 R U R2 U R2 U' R2 U2 R' U2 R(16/53) AUF: U2 (1/54)
Scramble 2: L2 F' L2 F L' D B D2 U2 L2 D L' R B' L2 U B2 R2 D2 R B2 D U2 B' F'
A RH OH ZZ approach
EOCross: L B' R' U D F R D R' D R' (11/11) BL slot: U2 R U R' L U L' (7/18) FL slot: L2 U2 L U L' U L2 (7/25) BR slot: U R' U' R U' R' U R (8/33) FR edge: R U R' (3/36) [It kills me not to do CLS for this case :(] CPLS setup: U2 (1/37) CPLS (O): y U2 z' U L' D2 L U' L' D2 z (8/45) 2GLL (T Ub2): z' U L2 U' L' U L' U' L U L U' L U L2 U' (15/60) AUF: L2 (1/61, or 57 with cancellations)
Scramble 3: L' R' B F U B2 D2 F2 L F2 D U' B' U D2 F2 B L2 R2 B U2 B' D2 U2 R
A CFOP approach Not working
Cross: x2 F D L2 U2 L F' (6/6) BL slot: F' U2 F L U' L' (6/12) FR slot: U R U2 R' U R' F R F' (9/21) BR slot: U' R' U2 R2 B' R' B (7/28) ELS setup: y U (1/29) ELS: R U' R' F' U2 F (6/35) CPLS (+): U' U' y R' U L' U' R U L (9/44) 2GLL (Pi Ub3): y' (y) R U2 R2 U' R' U R U' R' U' R' U' R' U2 R U2 (15/59, or 56 with cancellations)