Last Four Corners


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. L4C is and is not a method in the CxLL group, not because it has twice the number of cases as all the other methods of the group (42/84).
Usage:
 For a 2look last layer this is preceeded with LLEF (ELLCLL).
 FMC, aspecially linear FMC to solve the last layer corners but also normal FMC where a block + edge skeleton leaves 4 corners in diffrent layers, that are setup to one side and solved using L4C (not so effective but as a last resort).
 BLD, the sub group L3C is used by many freestylers and is also a part of the BeyerHardwick Method. E and X cases are double 2cyles, that can solve 4 corners in one go if the case is represented in the scramble or if it is easy to setup.
2look L4C
It is possible to solve most L4C in two looks using 3cycle commutators (see L3C). The method is to use the first 3cycle 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 3cycle. This will solve all cases but the pure twists with 4 corners wrong, that you can solve in two looks using only the Utwist and puzzle orientations, the other pure cases, T and U use the Utwist, L and S; cycle the twisted corners using any 3cycle and then the appropiate cycle to move them back.
Another way is to use COs that preserves edges but not corner permutation (see OCLLEPP) and then end the solve using CPLL. This will use lesser algs but more turns than the first approach.
L4C with a beginner method
Another approach is a beginner's method of last layer, permutation first:
U R U' L' U R' U' L  permute corners except frontrightup corner
R' D' R D  repeat  rotates corners. Must rotate only up face to switch which corner is rotated.
See also:
External links
L4C 
Pure CO  CPLL  L3C  L3C 1: 2twist , 3twist , 4twist  E cases  X cases  edit 
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, APLL a and b from all four angles, the two possible ways of EPLL and finally XPLL (that is HPLL + 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.
Recognition
For recognition the same systems that are used for CxLL works fine. You can find the CxLLs separated into the diffrent groups of the listing of the cases at this page, the CxLLs that have corners correctly permuted you can find in the group with pure CO and also in the XPLL group, the CxLLs with diagonal permutation are in the EPLL group and the CxLLs with adjacent permutation are in the APLL group (L3C/L3C 1twist).
Algorithms
Note that all of these algorithms are written in the Western notation, where a lowercase letter means a doublelayer turn and rotations are denoted by x, y, and z. (how to add algorithms) Click on an algorithm (not the camera icon) to watch an animation of it. 
The first algorithm given for each case is the optimal solution in half turn metric.
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 APLL (a and b), EPLL and HPLL (XPLL).
 One corner solved; Last Three Corners, 16 cases that have both orientation and permutation, these you can find at the L3C page.
L3C 1
One corner placed but twisted: 54 in the group, 18 are pure twists (see CO for the cases), 36 cases are having both orientation and permutation and of these 18 are mirror cases.
Two corners twisted
The names for the cases are first the actual orientation (U, T or L) followed by the orientation of the inverse case (the case you get if you apply the alg on a solved cube) and last is the direction for the permutation cycle (a or b).
The following four cases are mirror + inverses of the first so you only need '1 alg' for all. 
Mirror to the side and inverse in diagonal.  
ULa

ULb
 
LUa

LUb
 
The following four cases are mirror + inverses of the first so you only need '1 alg' for all. 
Mirror to the side and inverse in diagonal.  
TLa

TLb
 
LTa

LTb
 
For the following cases mirror and inverse are the same. 
Mirror/inverse to the side.  
TTa

TTb
 
UUa

UUb

Three corners twisted
The names here are following the images, the first letter is the orientation, either Sune (S) or Antisune (S), the second is the direction for the permutation cycle that is either a or b and in the parentesis is the location of the oriented corner that is one of three positions (BR, RF or FL).
The following four cases are mirror + inverses of the first so you only need '1 alg' for all. 
Mirror to the side and inverse in diagonal.  
Sa (RF)

Sb (RF)
 
Sa (BR)

Sb (FL)
 
The following four cases are mirror + inverses of the first so you only need '1 alg' for all. 
Mirror to the side and inverse in diagonal.  
Sa (BR)

Sb (FL)
 
Sa (RF)

Sb (RF)
 
For the following cases mirror and inverse are the same. 
Mirror/inverse to the side.  
Sa (FL)

Sb (BR)
 
Sa (FL)

Sb (BR)

Four corners twisted
The names for the cases are first orientation (H or pi) followed by the direction for the permutation cycle (a or b). In the parentesis is the orientation for the placed corner (+ or ) and for the pi cases also if this corner is on the U or the T side of the pi (For H it is only one situation).
The following four cases are mirror + inverses of the first so you only need '1 alg' for all. 
Mirror to the side and inverse in diagonal.  
Ha (+)

Hb ()
 
pia (T+)

pib (T)
 
The following four cases are mirror + inverses of the first so you only need '1 alg' for all. 
Mirror to the side and inverse in diagonal.  
Ha ()

Hb (+)
 
pia (U)

pib (U+)
 
For the following cases mirror and inverse are the same. 
Mirror/inverse to the side.  
pia (U+)

pib (U)
 
pia (T)

pib (T+)

Four corners wrong (E)
16 cases, two are EPLL, leaving 14.
Some cases are order 2 and some are order 6 but in practice most works as 2cycles if you add a y2. The rest of the order 6 cases may have a mirror or a inverse.
Four corners wrong (X)
8 to solve and one of these is XPLL, leaving 7, all solveable using only RU (2gen).
Tx is the inverse of Ux. 
These can be solved with optimal movecount with 2 conjugates (first alg in the lists).  
Ux

Tx
 
Lx is self inverting and self mirroring, only one case here. 
Any variation of triple Sune/anti/fat/mirror to solve fast (but not optimally).  
Lx

Sx and Sx are mirror cases. 
'SuneBruno' is the optimal alg.  
Sx

Sx
 
Hx

Pix

L4C 
Pure CO  CPLL  L3C  L3C 1: 2twist , 3twist , 4twist  E cases  X cases  edit 