Last Four Corners
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Last four corners, abbrevaited L4C (or 4LC), is a method that solves the last layer corners preserving all the rest, a sub group of ZBLL and ZZLL. For a 2-look last layer this is preceeded with LLEF (ELL first).
See also:
L4C Cases
Because edges are solved at this point you cannot AUF as in CxLL so the number of cases is quadrupled, but because you cannot have parity that is reduced again by factor 2. The orientations are the usual seven COs plus the solved case (CPLL). The permutations (with fix position orientations) that occure are permutation solved, A-PLL a and b from all four angles, the two possible ways of E-PLL and finally X-PLL (that is H-PLL + U2) giving a total of twelve possible permutations. This gives a total of 8 * 12 = 96 cases, but it is possible to reduce that a bit because of duplicates. The whole group is larger, corner orientations are for real 27 so you will have 27 * 12 = 324, so the chance of a skip is 1:324.
2-look L4C
It is possible to solve most L4C in two looks using 3-cycle commutators. The method is to use the first 3-cycle to solve one piece and at the same time move any placed but twisted corner out from position (to avoid the pure twists that uses long algs) and then follow it up with a second 3-cycle. This will solve all cases but the pure twists with 4 corners wrong, that you can solve in two looks using only the U-twist and puzzle orientations, the other pure cases, T and U use the U-twist, L and S; cycle the twisted corners using any 3-cycle and then the appropiate cycle to move them back.
Another way is to use COs that preserves edges but not corner permutation and then end the solve using CPLL. This will use lesser algs but more turns than the first approach.
Algorithms
Note that all of these algorithms are written in the Western notation, where a lowercase letter means a double-layer turn and rotations are denoted by x, y, and z.
Click on an algorithm (not the camera icon) to watch an animation of it.
Sub groups
Pure CO
See Corner Orientation for the seven cases of pure orientation and algorithms to solve them.
All corners oriented
Use the CPLLs for this, cases are A-PLL (a and b), E-PLL and H-PLL (X-PLL).
Mabye list the optimal algs for the cases here but with no images, for FMC usage (I'm trying to save a little on the size of the page ... and work creating images =)
L3C
Another sub group of L4C is Last Three Corners. Four of the cases are pure twists (U, L, S and -S) and two are the A-PLLs listed in the previous groups, leaving 16 that have both orientation and permutation, these you can find at the [[L3C|L3C page.
L3C 1-twist +
27 in the group, 9 are pure twists U, T, L, 3xS, H and 2xpi, 18 cases are having both orientation and permutation.
Two corners twisted
1+ Ua
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1+ Ub
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1+ Ta
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1+ Tb
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1+ La
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1+ Lb
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Three corners twisted
1+ S1a
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1+ S1b
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1+ S2a
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1+ S2b
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1+ S3a
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1+ S3b
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Four corners twisted
1+ Ha
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1+ Hb
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1+ pi1a
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1+ pi1b
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1+ pi2a
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1+ pi2b
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L3C 1-twist -
The 18 mirror cases of the previous group.
Two corners twisted
1- Ua
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1- Ub
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1- Ta
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1- Tb
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1- La
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1- Lb
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Three corners twisted
1- S1a
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1- S1b
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1- S2a
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1- S2b
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1- S3a
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1- S3b
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Four corners twisted
1- Ha
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1- Hb
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1- pi1a
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1- pi1b
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1- pi2a
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1- pi2b
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Four corners wrong (E)
16 cases, two are E-PLL, leaving 14.
U E (a)
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U E (b)
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T E (a)
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T E (b)
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L E (a)
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L E (b)
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S E (a)
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S E (b)
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-S E (a)
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-S E (b)
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H E (a)This case can be solved optimally by simply doing OLL 45 three times.
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H E (b)
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pi E (a)
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pi E (b)
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Four corners wrong (X)
8 to solve and one of these is X-PLL, leaving 7, all solveable using only RU (2-gen).
U X and its inverse T X can be solved with optimal movecount with 2 conjugates.
S X
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-S X
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H X
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Pi X
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