Full PLL Algorithms: All 21 Cases, Each With a Picture

What is full PLL?

PLL (Permutation of the Last Layer) is the last step of CFOP. When it starts, the whole top face is already yellow after OLL. All that is left is to swap the top layer pieces around so the colours on the side faces match the centres.

2-look PLL does this in two goes: swap the corners, then swap the edges. Full PLL does it in one go, with its own algorithm for each arrangement. According to the SpeedSolving Wiki there are 21 of these arrangements, and the step was proposed by Hans Dockhorn, Anneke Treep and Jessica Fridrich (SpeedSolving Wiki). Ruwix also lists exactly 21 cases (Ruwix).

The picture below shows the case you meet most often in 2-look PLL, the T-perm, which is also one of the 21:

When should you move on to full PLL?

J Perm suggests going for full PLL when you feel ready, usually somewhere around a sub-30-second average (J Perm). Before that point, most of your time goes on the cross and F2L, so extra top layer algorithms won't help much yet.

You don't need all 21 at once. Each new algorithm turns a "look twice" case into a "look once" case. For the cases you don't know yet, just keep solving them with 2-look PLL as before.

How to read the pictures and notation

• Yellow on top, green in front. In the pictures the right face is orange. The left face (hidden) is red and the back face (hidden) is blue.
• Each picture shows two cubes: on the left is the case you have, on the right is the cube after the algorithm. Only the top layer is coloured.
• The pictures only show the front and right faces. Under each case there is a line giving the colours of the top row of the back and left faces, read from left to right as you look straight at that face.
• Before doing the algorithm, turn the top face (U, U' or U2) so your cube looks like the left-hand picture.

Extra notation you need, besides the moves R, L, U, D, F, B:

• M: turn the middle slice (between L and R) the same way as L.
• x: turn the whole cube the same way as R. x' is the other way.
• y: turn the whole cube the same way as U.

The pictures for algorithms with M, x or y are drawn with an equivalent algorithm that only uses the outer faces. The case in the picture is exactly the case for the bold algorithm. If the notation is new to you, see Rubik's Cube notation for beginners.

Three words for spotting cases

• Headlights: on one side face, the stickers of the two corners are the same colour. Those two corners are already in the right place compared with each other.
• Pair: a corner and the edge next to it have the same colour on one side face, making two matching stickers side by side.
• Bar: all three stickers in the top row of one side face are the same colour.

Counting the faces with headlights is the quickest way to tell which group a case is in:

• Headlights on all four faces: the corners are done, so only the edges need to swap. Group 1.
• Headlights on all four faces, and the edges are right too: the cube is done; just turn U.
• No face has headlights, but every edge matches its centre: E-perm (group 2).
• Headlights on exactly one face: two neighbouring corners need to swap (the A-perms are in group 2, the rest are in group 3).
• No face has headlights: two diagonal corners need to swap. Group 4.

The split into three groups, "edges only", "corners only" and "both corners and edges", follows the SpeedSolving Wiki. In this post the last group is split again in two by which corners need to swap: neighbouring or diagonal.

Group 1: edges only (Ua, Ub, H, Z)

All four corners are already right, so all four faces have headlights. You already know all four of these algorithms from 2-look PLL.

Ua-perm

One face already has a bar: put it at the back. The front edge has the colour of the right face (orange).

Back: blue - blue - blue. Left: red - green - red.

R U' R U R U R U' R' U' R2

Ub-perm

There is still a bar at the back, but the front edge has the colour of the left face (red).

Back: blue - blue - blue. Left: red - orange - red.

R2 U R U R' U' R' U' R' U R'

H-perm

No bar. Each edge has the colour of the opposite face: the front face has a blue edge and the right face has a red edge. You can hold the cube any way.

Back: blue - green - blue. Left: red - orange - red.

M2 U M2 U2 M2 U M2

Ruwix uses this exact algorithm for H (Ruwix).

Z-perm

No bar. Each edge has the colour of a neighbouring face. Hold the cube so the front edge and the right edge are the two that swap: the front edge is orange and the right edge is green.

Back: blue - red - blue. Left: red - blue - red.

M2 U M2 U M' U2 M2 U2 M' U2

Group 2: corners only (Aa, Ab, E)

In this group every edge already matches the centre of its face. Only the corners need to swap. All three algorithms start by turning the whole cube (x or x') and end by turning it back; the SpeedSolving Wiki writes them the same way.

Aa-perm

Headlights on the back face. The other three corners need to cycle around. The front face has a green pair on the left.

Back: orange - blue - orange. Left: blue - red - red.

x R' U R' D2 R U' R' D2 R2 x'

Ab-perm

Headlights on the right face (two blue corners with an orange edge in the middle). The front face also has a green pair on the left.

Back: red - blue - green. Left: orange - red - red.

x R2 D2 R U R' D2 R U' R x'

Aa and Ab are the two directions of the same three-corner cycle. If you pick the wrong one, you will see another A-perm: just do one more, matching its picture.

E-perm

No face has headlights, but every edge is right. On each side face, the two corners are different colours from each other and from the edge in the middle.

Back: orange - blue - red. Left: green - red - blue.

x' R U' R' D R U R' D' R U R' D R U' R' D' x

The E-perm is long, but between the two cube turns there are four chunks of four moves that follow the same pattern: R, U, R', D. The only thing that changes from chunk to chunk is the direction of U and D.

Group 3: swap two neighbouring corners (T, F, Ja, Jb, Ra, Rb, Ga, Gb, Gc, Gd)

This is the biggest group. Every case has headlights on exactly one face, because two neighbouring corners need to swap. In the pictures, most cases are held with the headlights on the left face; to tell them apart, look at the pairs and bars on the other faces.

T-perm

Headlights on the left face, with an orange edge in the middle. The front face has a green pair on the left, and the back face has a blue pair on the right. The right face has no pair.

Back: orange - blue - blue. Left: red - orange - red.

R U R' U' R' F R2 U' R' U' R U R' F'

F-perm

A bar on the left face. The front face has a blue edge between two corners of different colours. No other face has a pair.

Back: orange - green - blue. Left: red - red - red.

R' U' F' R U R' U' R' F R2 U' R' U' R U R' U R

The F-perm is a T-perm "wrapped" in a few extra moves: three moves R' U' F', then the first 13 moves of the T-perm, then U R. If you already know T, F will come to you very quickly.

Ja-perm

A bar on the front face (red). The other three faces each have a pair: the right face has a green pair on the left.

Back: blue - blue - green. Left: orange - orange - blue.

R' U L' U2 R U' R' U2 R L

This algorithm has no cube turns. Ruwix gives it for the J case (Ruwix).

Jb-perm

A bar on the left face (blue). The front face has a green pair on the right, and the right face has a red pair on the right.

Back: green - orange - orange. Left: blue - blue - blue.

R U R' F' R U R' U' R' F R2 U' R'

The second part of Jb, from the fifth move to the end, is exactly the first 9 moves of the T-perm.

Ra-perm

Headlights on the left face, with an orange edge in the middle. The front face has a red pair on the left. The right and back faces have no pair.

Back: green - blue - orange. Left: blue - orange - blue.

R U' R' U' R U R D R' U' R D' R' U2 R'

Rb-perm

Headlights on the left face, with an orange edge in the middle. The back face has a red pair on the right. The front and right faces have no pair.

Back: blue - red - red. Left: green - orange - green.

R2 F R U R U' R' F' R U2 R' U2 R

Telling the four G-perms apart

A G-perm swaps two neighbouring corners and cycles three edges at the same time. There are four versions, and they are the hardest cases in PLL to tell apart. All four pictures below have headlights on the left face. Look for the only pair on the other three faces:

• Ga: an orange pair on the right of the front face.
• Gb: a green pair on the right of the right face.
• Gc: an orange pair on the left of the back face.
• Gd: a blue pair on the left of the right face.

All four algorithms have D moves, because they have to swap both corners and edges without breaking the two bottom layers.

Ga-perm

Back: orange - green - blue. Left: red - blue - red.

R2 U R' U R' U' R U' R2 U' D R' U R D'

Gb-perm

Back: orange - red - blue. Left: red - orange - red.

R' U' R U D' R2 U R' U R U' R U' R2 D

Gc-perm

Back: orange - orange - blue. Left: red - green - red.

R2 U' R U' R U R' U R2 U D' R U' R' D

Gd-perm

Back: orange - green - blue. Left: red - orange - red.

R U R' U' D R2 U' R U' R' U R' U R2 D'

Notice the pairs: Ga and Gb are inverses of each other, and so are Gc and Gd. For example, take Ga, read it backwards from the end and reverse each move, and you get an algorithm for Gb.

Group 4: swap two diagonal corners (V, Y, Na, Nb)

No face has headlights, and some edges are out of place (if every edge is right, it is the E-perm from group 2).

Y-perm

The front face has a green pair on the left, and the right face has an orange pair on the right. The two pairs sit at two diagonal corners of the top face.

Back: blue - red - green. Left: orange - blue - red.

F R U' R' U' R U R' F' R U R' U' R' F R F'

V-perm

The front face has a green pair on the left, and the left face has a red pair on the right. The two pairs meet at the front-left corner, making a block of three pieces when you look from above.

Back: blue - orange - green. Left: orange - red - red.

R' U R' U' y R' F' R2 U' R' U R' F R F

The y move is in the middle of the algorithm: after R' U R' U', turn the whole cube the same way as U, then carry on. Ruwix writes the first part with a d' move (turning the bottom two layers), which gives the same result (Ruwix).

Na-perm

All four faces have a pair, and every pair is on the right of its face.

Back: green - blue - blue. Left: red - orange - orange.

R U R' U R U R' F' R U R' U' R' F R2 U' R' U2 R U' R'

Nb-perm

All four faces have a pair, and every pair is on the left of its face.

Back: blue - blue - green. Left: orange - orange - red.

R' U R U' R' F' U' F R U R' F R' F' R U' R

Na and Nb are rare. It is fine to leave them until last.

Which cases come up most often?

We counted every possible top layer arrangement after OLL (288 of them) on the simulator. Each case takes a different share, depending on how many angles it looks the same from:

So the 16 common cases make up almost 90% of solves. H, Na and Nb each come up only about once in every 72 solves.

A suggested order

The order below is our suggestion, based on the table above and on which algorithms look like ones you already know:

1. The six algorithms from 2-look PLL: T, Y, Ua, Ub, H, Z. You already have these.
2. Aa, Ab: short, only 9 moves inside the x ... x' frame.
3. Jb, F: very close to the T-perm.
4. Ja, Ra, Rb: they come up often and have no cube turns.
5. V: comes up as often as Y, and it is also a diagonal corner swap.
6. Ga, Gb, Gc, Gd: one inverse pair at a time.
7. E, Na, Nb: the rarest ones, so save them for last.

Ruwix suggests remembering algorithms by how your hands move, not just by the notation (Ruwix). How to practise a new algorithm:

• Starting from a solved cube, do the algorithm backwards (read it from the end and reverse each move) to set up the case. Then solve it again with the algorithm.
• Say the name of the case before you do it, so you practise spotting the case and not just your hand movements.
• Once you know a few, solve the whole cube and time yourself with the 3x3 timer.

Common mistakes

• Forgetting to turn U first. Each algorithm only solves the case when your cube looks like the left-hand picture.
• Mixing up Ga with Gc, or Gb with Gd. Don't look at the headlights; look for the only pair on the other three faces.
• Mixing up Na and Nb. Look at any face: a pair on the right means Na, a pair on the left means Nb.
• Doing x or y the wrong way. x turns the same way as R, and y turns the same way as U.
• The cube is still one turn off after PLL. Just turn U, U' or U2.

What to do next

Once you have all 21 PLL algorithms, you have every last layer algorithm in CFOP. From here, most of your speed comes from seeing what comes next before you turn. Go back over the whole series: how to do the cross, the 41 F2L cases, the 57 OLL algorithms, and the overall path in What is CFOP.

With the EasyRubic app, you can scan your real cube with the camera and see the solution step by step on a 3D cube. Get EasyRubic to practise each PLL case on your own cube.

Video

Want to see it in motion? The short video below from the TuấnRubik Cube channel on YouTube (in Vietnamese) shows more of what this post covers. The video belongs to that channel and is embedded straight from YouTube.

Sources

• PLL (SpeedSolving.com Wiki)
• Permute the Last Layer (PLL) – Ruwix
• CFOP Method – J Perm