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These are the 21 permutation cases for the last layer and the algorithms I use for them.  These algorithms appear EXACTLY as I perform them when I am solving the last layer, in speedcubing notation with rotations included in the algorithm.  It should be noted that these are the algorithms that I find easiest to perform.  However, you may find other algorithms better-suited for your own hands, so it is recommended to try many different algorithms for the same situation to find which one works best for your own style of cubing.
I have made fast and slow videos to help demonstrate how I perform each algorithm.  The fast algorithms probably aren't of much help, but I like them because they give a good indication of how fast an algorithm can be performed.
In each diagram, the edges that are being swapped or moved are denoted by the red arrows, while the corners that are being swapped are moved are shown with blue arrows.
If you do not understand any of the notation, please visit the notation page.
At the very bottom of the page, I have listed some links to other pages with OLL algorithms.
For a printable page of these algorithms, visit my printable page.

Corners Only
In cases #01 and #02, I first AUF (adjust the U-face) until the "corner block" is permuted.  By corner block, I mean that there is a corner and the two edges adjacent to it belong adjacent to it.  Since this is very similar to case #18, i check to see that the two stickers next to the corner block are the same color (if they are not, then this is case #18).  To distinguish between the two cases, I look at both sides of the corner block.  I rotate the cube so that whichever of those sides shows the sticker of a corner that belongs opposite along the diagonal of where it is located.  I use that as an indicator for which algorithm to use.  For example, if there is a solved blue-red corner block, and the rightmost sticker on the red side is orange, then I know I have Case #01.  Case #03 is obvious to recognize because there are no 1x1x2 (or larger) blocks.  Thus, I just AUF to permute any edge (the other three will be permuted by default), and rotate the cube to match the diagram below.
# Diagram Time Algorithm Comments Video
01 1.300 (Lw' U R') D2 (R U' R') D2 R2 This is a basic corner 3-cycle.  It is one of my favorite and fastest algorithms.  The algorithm is a lot simpler than it looks. slow
fast
02 1.533 (Rw U' L) D2 (L' U L) D2 L2 This is just the mirror of #01. slow (old)
fast
03 1.933 x' (R U') (R' D) (R U R') Uw'2 (R' U) (R D) (R' U' R) The last five turns are usually extremely fast.  This is easy to recognize because other than algorithms #04 and #05, it is the only one that does not have any 1x1x2 blocks. slow
fast

Edges Only
In these cases, I first AUF until the corners are all permuted.  Cases #04 and #05 are very easy to recognize because there are no solved 1x1x3 blocks (hence, no solved faces).  Noticing that two adjacent edges need to be swapped indicates to perform case #04 and if two opposite edges need to be swapped, the case is #05.  I think it is important to be able to recognize Cases #06 and #07 from all angles and be able to execute them using any grip.
# Diagram Time Algorithm Comments Video
04 1.800 (M'2 U) (M'2 U) (M' U2) (M'2 U2) (M' U2) This is a very fast algorithm that Gilles van den Peereboom showed me at WC2005.  The last U2 is not necessary if you account for it before the algorithm. slow
fast
05 1.133 (M'2 U) (M'2 U2) (M'2 U) M'2 This is extremely easy to recognize and can be performed VERY quickly.  The M'2 is actually performed as (M'M') with rapid pushing at the back face of the M layer with the middle and then ring fingers.  Some people call this the "Bob Burton H-perm" because my discovery of this finger trick enabled me to perform this algorithm at insane speeds (at best under one second). slow
fast
06 1.300 (R U' R U) (R U) (R U') (R' U' R2) This is just a simple 3-edge cycle.  It is almost as faster than the corner cycles. slow
fast
07 1.133 (R2 U) (R U R' U') (R' U') (R' U R') This is the inverse of #06.  I place my hands slightly differently for this algorithm. slow
fast

Swapping Two Adjacent Corners & Two Edges
For cases #08 and #09, I AUF to solve a face and then rotate the cube such that the solved face is in front.  Then, it is easy to determine which of the two cases you have.  Case #10 is unlike any other because it has two 1x1x2 blocks directly across from each other.  I recognize it by first AUF such that the blocks are solved and then rotate the cube such that the blocks are on the left side.  Cases #11 and #12 are a bit trickier than most other cases to recognize.  I first AUF to solve the 1x1x2 block.  Then, I rotate the cube so that the two unsolved corners are in the back and then determine which of the two cases I have.  For case #13, first AUF to solve a face.  Then, since the edge opposite that face is correct, it is immediately obvious that the cube must be rotated so that the solved face is at the back so that the algorithm can be performed.
# Diagram Time Algorithm Comments Video
08 1.800 (R' U L') U2 (R U' R') U2 ([L R] U') This situation comes up somewhat often and is quite easy to recognize.  I perform the R of the Ra a split second after I start the L so that I can immediately perform the U' when the L face has been moved to where it belongs. slow
fast
09 1.400 (R U R' F') (R U R' U') (R' F) (R2 U') (R' U') I have to thank Quinn Lewis for this alg.  It rocks my world.  It is the same as PLL #10 with the last four moves instead performed at the beginning. slow
fast
10 1.500 (R U R' U') (R' F) (R2 U') (R' U' R U) (R' F') This is the "T" permuation.  It is long but definitely very fast and easy.  It is also very easy to recognize.  It can be performed in almost one swift motion without any readjusting of the fingers.  Note that it is a combination of two easy orientations. slow
fast
11 1.767 (R' U2) (R U2) (R' F R U R' U') (R' F' R2 U') Quinn Lewis showed me an effective algorithm for this case that I have fallen in love with.  It is about twice as fast as the algorithm I was previously using. slow
fast
12 1.867 (L U'2) (L' U'2) (L F' L' U' L U) (L F L'2 U) This is just the mirror of Case #11. slow
fast
13 2.533 (R U') (R' U R'2) y (R U R' U' F' Dw) (R'2 F R F') I am going to swith to Stefan Pochmann's algorith for this case. slow
fast

Cycling Three Corners & Three Edges
Though these look the trickiest to recognize, they are actually quite simple.  I first AUF to solve the 1x1x2 block.  Then, I rotate the cube such that the two corners that share the same color on the same face are on the left side.  Then, based on whether the block is at the back, front, far part of the right, or close part of the right, I know whether to apply #14, #15, #16, or #17, respectively.
# Diagram Time Algorithm Comments Video
14 2.233 (R'2 Uw' R U') (R U R' Uw R2) y (R U' R') This is fairly easy to perform at high speeds, even though it looks the most confusing.  Algorithms #14-#17 are all performed somewhat similarly because they have some overlapping moves. slow
fast
15 1.867 (R'2 Uw) (R' U R' U' R Uw') R'2 y' (R' U R) Ron showed me a nice modification to this algorithm to make it flow a lot nicer.  It is quite easy to perform with a little practice. slow
fast
16 2.167 (R' U' R) y (R'2 Uw R' U) (R U' R Uw' R'2) This is the inverse of #15.  Note how similar they look.  I perform this one almost exactly the same way. slow
fast
17 2.100 (R U R') y' (R'2 Uw' R U') (R' U R' Uw R2) This is just the inverse of #14.  I execute it very similarly because most of the moves overlap in the same manner. slow
fast

Permutations Of Two Diagonal Corners & Two Edges
Case #18 is recognized in the same manner as #01 and #02, except both sides of the corner block will have the opposite color in the corners.  Then, just rotate the cube to put the corner-block at the front-left side.  In cases #19 and #20, there are two sets of 1x1x2 blocks where the blocks are opposite each other, but "offset" (ie - not directly across from each other).  I first AUF to correctly place either of the two sets of blocks and then rotate the cube so that the ULB and URF corners are correctly placed.  Then, depending on whether the two edges that need to be swapped from front to back or left to right, it is clear whether I have case #19 or #20.  Finally, case #21.  This is one of the easiest to recognize.  First, AUF to correctly place the two perpendicular 1x1x2 blocks.  Since no other PLL case has this, rotate the cube and perform the algorithm.
# Diagram Time Algorithm Comments Video
18 1.767 (R' U R' Dw') x (Lw' U R' U') (Lw R U') (R' U R U) I perform the x rotation as I finish the first group of moves.  In the third group of moves, I start the R turn right after I start the l turn. slow
fast
19 2.700 (R U') (R' U) (Lw U) (F U') (R' F') (R U' R U) (Lw' U R') I got this algorith from Stefan Pochmann.  It only took me a couple minutes to get used to it. slow
fast
20 3.033 (L' U) (L U') (Rw' U') (F' U) (L F) (L' U L' U') (Rw U' L) This alg blows. slow
fast
21 1.800 (F R U') (R' U' R U) (R' F') (R U R' U') (R' F R F') This is the "Y" permutation.  It is very quick and can be performed without any adjustments of where the fingers are.  It is just a combination of two quick orientations. slow
fast

Other Pages With PLL Algorithms
Jess Bonde
Ron Van Bruchem
Loïc Frémont
Jessica Fridrich
Chris Hardwick
Dan Harris
Yuki Hayashi
Peter Jansen
Katsuyuki Konishi
Shotaro "Macky" Makisumi
Jon Morris
Dennis Nilsson
Ross Palmer
Richard Patterson
Lars Vandenbergh