Moves for doing the last layer of Rubik's Cube

What follows is an explanation of why I wrote this page when there are so many solutions on the net already. If you only want the solution, then skip on to here.

#### Introduction

I could always solve the first two layers of the cube just by moving things around, utilising the fact that changes made to what hasn't been done yet don't matter. As the cube progresses though it gets harder to do this, since it is difficult to move a piece without affecting something that has already been done. For the last face I always had to resort to set moves, which I originally obtained from reading the now famous booklet by David Singmaster. To save me hunting through the booklet for the moves I needed, I drew up a crib sheet on Isometric graph paper.

Years later, having forgotten all about the cube, a friend had found his and had got it to the state where just the last layer needed to be done. He too had originally learnt from "The Bible" but no longer had it and asked in the Rubik's conference on Cix if anyone had a copy or knew the moves. Unfortunately a search of my many "to be sorted" boxes did not reveal my copy, but I did find my crib sheet. A search of the Internet finds lots of references to David Singmaster's work, but no where is there a transcript of this famous booklet. Neither could I find anyone who had documented the moves we wanted for the last layer. So in desperation I resolved to work out what my crib sheet meant. The moves were plain enough but it wasn't clear what they did.

My trawl on the Internet had turned up the excellent Cube Explorer program by H. Kociemba and using this I was able to revive my memory. Having done this I decided to document it so that I wouldn't forget it again.

#### Nomenclature

If you already know the conventions you can skip on to the moves themselves.

The conventions are as in David Singmaster's authoritative guide. A cube has centre pieces, edge pieces and corners. The centre pieces are fixed relative to each other and cannot move their positions, just rotate. The edge pieces and corner pieces can migrate around the cube. The edge pieces can flip and the corner pieces can twist. A cube can be coloured in different ways and may not be the same as shown here. Because of this the colours are not used to refer to the cube, instead the faces are named F (Front), R (Right), L (Left), U (Up), D (Down), B (Back). The assignment of faces to colours is arbitrary, but you use the colours to keep track of the orientation. The cube, or at least your mental image of it, must be kept in the same orientation, as defined by the centre pieces throughout.

To refer to the pieces they are named by the position they occupy, i.e. which faces they are part of. So an edge piece has a two letter name; FU, RF, FL, etc. The ordering of the letters is immaterial, FR is the same piece as RF. In describing how the pieces move however, the order, though arbitrary, does matter. The corner pieces have three letter names such us URB, URF, etc. Note that only certain combinations are valid, UD and FBL are not valid names for pieces.

The moves are described by the face that is twisted. A clockwise turn is represented by the single letter of the face name. An anti-clockwise move is represented by the face name with ' added. FF is shortened to F2. F3 is the same as F' and so is not used. F4 does nothing in effect since the cube is back where it started before the move. If a series of moves is repeated it is placed in round brackets and a super-script indicates the number of repeats.

The effects of moves are shown as a list in round brackets of the pieces affected and the positions they move to. If two pieces swap they would be shown as (FU,UR). This means that the piece in the FU position moves to the RU position, but is flipped, i.e. the F of FU moves to the U of UR and the U of FU moves to the R of UR, viz the order matters. This move could also have been written as (UF,RU). Note that the second piece moves to the position of the first under the same rules. Often, pieces do not swap but move in a cycle of three. An example of this is (FU,UB,UR), the cycle is always read from left to right. The same convention is used for showing how corners move e.g. (FUR,UFL)(BUR,UBL), here the FUR and UFL pieces swap places and the BUR and UBL pieces swap places.

#### The moves

Orient the cube so that the two completed layers are underneath, and the layer to be done forms the U face. In the diagrams below the bottom (orange) layer, or D face, remains unchanged throughout (except procedure E) and so is not shown. The B and L faces which would be hidden in the perspective view have been 'hinged' out to show them, (technique cribbed from H. Kociemba). In showing the effects of these procedures it is possible to depict it with two diagrams; 'before' and 'after' images. Whether you prefer to visualise the move as restoring a scrambled cube to the completed position or how the move would affect a clean cube is a question of how it is easier for you to visualise it. I have shown both in the diagrams below. Either use the first two cubes for the first method, or the last two cubes for the second.

Start by getting the top edges the right way up (correct colour on top). Ignore the corners at this stage. They will get moved in these procedures but being a male I can only concentrate on one thing at a time. The corners will be sorted out later. If some of your corners are already right, and you want a minimum move solution, then this method is not for you.

A : FURU'R'F' = (FU,UB,UR) (ignoring corners)
A': FRUR'U'F' = (FU,UR,UB) (ignoring corners)

Next get them in the right positions. You must have at least one edge that is in the right position by turning the U face round until something lines up. If you are lucky you can then use either the B or the B' procedure to get the others right in one go.

B : (U²R²)³B'UB(U²R²)³B'U'B = (FU,RU,BU)
B': B'UB(U²R²)³B'U'B(U²R²)³ = (FU,BU,RU)

Now put the top corner pieces in their right place using either of these next two procedures.

This procedure swaps and twists all four corners in pairs.

C : F(URU'R')³F' = (FUR,UFL)(BUR,UBL)
C': F(RUR'U')³F' = (FUR,UFL)(BUR,UBL)

This procedure cycles three corners. If you already have one corner right then you should be able to get the others right in one move by noting whether to use Procedure D or the inverse procedure D'.

D : URU'L'UR'U'L = (FUL,RBU,UBL)
D': L'URU'LUR'U' = (FUL,UBL,RBU)

Finally twist the corners in to the correct orientation. This move twists one corner but leaves the rest of the top face undisturbed. It does however scramble the bottom, but don't worry about it, doing the inverse moves in the opposite order will undo the damage. Having twisted one corner move the top face to a new position and apply the inverse move which will apply the opposite twist to the new FUR corner and unscramble, i.e. restore, the bottom two layers (phew!).

E : R'DRFDF' = (FUR)+ {lower layers scrambled}
E': FD'F'R'D'R = (FUR)- {lower layers unscrambled}

#### Endpiece

There are many different ways to solve the cube and the method that is best for you depends on how your mental processes work, and how you visualise the cube, and what is happening to it, during the process. It is usually a trade off between a lot of different procedures for different circumstances for a quick solution, or a few simple procedures that are applied over and over again until the solution eventually drops out. The method I have presented here is definitely one of the latter ones. There are people who don't use crib procedures at all but just twist away and can do it 90 seconds or so.

I will resist the temptation to make a page of Cube links since much of the stuff out there is on personal pages which are apt to move without leaving a re-direction link, or even just disappearing. However the official site will hopefully be around for a bit.

Happy cubing.