L5E

L5E, Last five edges, is an experimental method that solves five edges, the four in U and one in the D layer, preserving all other pieces.
ELL is a sub group of L5E, all ELL's may be solved in two steps using this method.
Contents
Description
There are two main subparts. First the edges are oriented using M' U M like in Roux; there are 5 cases for this, and the worst case in Roux (6 edges) is impossible. Then the edges must be permuted; you can place FD during orientation or right after using [U] M' U2 M (named 'P') and end with EPLL, or just permute all five edges in one step.
L5E can be useful for Corners First if the Eslice, centres and three FL edges are solved after the corners. It is also useful in any method that starts with columns, such as one which does the F2L pairs first and then finishes with CMLL, it is an alternative to the last step of Roux if the centres are placed together with the BD edge at first, also for any LBL style, cross minus one edge at first and then the LL corners have been finished with CLL or CMLL or the same for Fridrich with 2look OLL, corners first, then L5E orientation, place FD using P if it was not solved and end it all in PLL as usally.
Zipper L5E/L5EFR
Zipper L5E or L5EFR is used in the Zipper Method, specifically in Zipperb. The difference between this variant and standard L5E is that instead of the FD edge, the FR edge is permuted. This approach, however, cannot be solved 2Gen (with only M and U moves) like standard L5E.
L5EP

Last 5 Edges Permute, often abbreviated L5EP, is a 3x3 subset that permutes the 4 edges on the last layer and an edge from another layer while simultaneously preserving all other pieces.
The five affected edges must be correctly oriented for L5EP to work. L5EP is often used in methods like PetrusW and Portico, when there are only five edges left to solve on the cube.
There are 16 algorithms for L5EP, 10 excluding mirrors and 7 excluding mirrors and EPLL.
Variants
Although it is generally agreed upon that L5EP permutes the four U layer edges, the location of the fifth edge is not always clear. Because of that, multiple variants exist:
Standard L5EP (L5EPU/D): This is the original proposal by Mike Bennet where the UF, UR, UB, and UL edges and an edge on the D layer are permuted simultaneously. It originated as an alternate strategy for the Roux method's last six edges step.[1]. It was independently proposed again in 2006 by James Straughan as part of the MI1 method and again in 2008 by Jason Baum[2]. That same year, Kenneth Gustavsson developed a naming scheme and James Straughan developed a way to represent the cases in images[3]. Any case can be simplified to an EPLL by first permuting the D layer edge. Intuitive L5EP can also be used, but it is recommended to learn algorithmic L5EP due to the faster execution and low number of algorithms.
EHKPLL: EHKPLL is a subset of HKPLL which was invented for Hawaiian Kociemba. Here, the fifth permuted edge is the DR edge.
L2L4 / Zipper L5EP (L5EPU/E): This variant was first used in Duncan Dicks' L2L4 method.[4] In Duncan Dicks' variant, the F layer is the last layer and the fifth edge is on the S layer at UR. This same variant is also used in the Zipper Method and permutes the U layer edges and either the FR or BR edge.
See also
External links
Intermediate system
Intuitive
This style is not completely intuitive, because it is still necessary to know EPLL. First orient all unoriented edges, then place the FD edge to its correct position and finally end the solve in EPLL. The necessary intuitive steps are:
 P = M' U2 M
 O = M' U M
 O' = M' U' M
Using that plus some U turns you can solve any case, even all ELL case. For example, O U P = (M' U M) U (M' U2 M) is a sample algorithm which solves orientation, then places FD. There is a 1 in 12 chance that EPLL is skipped; otherwise there is one algorithm left.
There is another notation which can make it shorter to write algorithms:
 F(moves) = F2 (moves) F2
It is important here that the moves inside the parentheses keep the corners solved, so that they will still be correct after this algorithm. An example is the Uperm, F(U P U) = F2 (U M' U2 M U) F2. F(O U') is a nice way to solve orientation and place FD all at once if the FD edge is in the middle of three unoriented edges in LL.
Using algorithms
The intuitive style lets you find and understand the orientation algorithms yourself. Another approach is to learn the algorithms instead of finding them; the cases are:
More algs are at the L5EOP page
Sometimes this will also solve the FD edge; the images show where the FD edge must be for the algorithm to solve it, but otherwise you should simply treat the yellow sticker as white. If you learn these positions, it can also be useful to learn the mirror algorithm for the case where FD is on the opposite side from the image. The easiest example is to use O' instead of O for the case with three flipped edges on U. Just like the intuitive step, if FD is not solved use P, and either way finish with EPLL.
Semiadvanced system
L5EOP is an alternative to L5E, that always solves the FD edge while orienting ('P'lace). Then the last step will always be EPLL.
FD placement + EPLL (L5EP)
Another possibility for improving the intermediate style is to permutate the last five edges in one step. This only requires 16 algorithms in total, so counting the 5 orientations creates a system that solves L5E in two short steps. Even if you know this, it is still useful to place FD without extra moves if possible, since EPLL recognition is very fast.
We will introduce the following function:
 f(moves) = f2 (moves) [U] f2 ... This is the same as the F() function above, but with double layer f turns. Note that sometimes the adjusting U turns are not mentioned.
Here are the algorithms for permutation of all 5 edges; make sure to use AUF to place the FD edge at the position it has in the image.

 

 

 

 

 



 

 



Advanced system
The most advanced system is to solve all edges in one step. Solving the last five edges in a single step contains 245 algorithms and reduces the number of looks and the movecount. There are a few variants of L5E. One solves four edges of the U layer and one on the D layer. The first known developed and published method to use this single step L5E is the Nautilus method. Another L5E variant is one where the four U layer edges and an edge in the E layer are solved. This was first proposed for the Zipper method. A final L5E variant that has been proposed occasionally is one which solves three edges of the U layer and two edges on the D layer. This can be used in corners first and related methods such as Waterman, Roux, and LMCF.