How to solve rubik's cube 3x3x3

Terminology

When describing the solution for the 2nd and 3rd layers, standard cube notation will be used. Here's what you need to know to read it:

F = front face

B = back face

R = right face

L = left face

U = up face

D = down face

In addition to a letter, each move may be accompanied by an apostrophe or the number two:

• A letter by itself means turn that face 90 degrees clockwise (eg. F).
• A letter followed by an apostrophe means turn that face 90 degrees anti-clockwise (eg. F').
• A letter followed by the number 2 means turn that face 180 degrees (direction is irrelevant), (eg. F2).

So R U' L2 is shorthand for "turn the right face 90 degrees clockwise, then turn the up face 90 degrees anti-clockwise, then turn the left face 180 degrees". When thinking whether to turn clockwise/anti-clockwise, imagine that you are looking directly at the particular face you are turning.
For each algorithm, the notation is written with the assumption that the core of the cube remains fixed throughout the whole algorithm, and the faces just turn around it. This means that you also need to know how to position the cube to start the algorithm.

The Solution

The First Layer

The first layer is solved in two stages:

1. Form the cross
2. Insert the 4 first layer corners (each corner is inserted individually)

I believe that the first layer should be done intuitively. You need to understand it and solve it without learning algorithms. Until you can do this, I wouldn't bother attempting the rest of the cube! So, spend some time playing with the cube and familiarising yourself with how to move the pieces around the cube.
Now, here are some tips to get you started.

The Cross

I prefer to start with the white cross because I find white easier to quickly identify on a completely scrambled cube, however, you can use any colour.
There are 4 edge pieces with white (ie. the 4 arms of the cross) which have specific positions. You can't put any white edge piece in an arm of the cross because the other colour on the edge cubie must match up with it's centre on the middle layer.

Here is a pic of what a correctly formed cross looks like (grey denotes cubies that are irrelevant to the cross). Note that the white/red edge cubie matches up with the white centre and the red centre. Ditto re the white/blue cubie.

Here's a pic on an incorrectly formed cross. Looking at the white face we do indeed see a white cross, however the white/red edge cubie does not match up with the red centre. Ditto re the white/blue cubie. This is bad!

For a detailed explanation of the cross, check out Dan Harris' Solving the Cross page.

The First Layer Corners

Once you have completed the cross, completing the first layer requires inserting each of the 4 corners in separately. The first thing to do is examine your cube and locate all of the top layer edge pieces - they will be sitting in either the first layer or the last layer. Inserting the first layer corners should be done intuitively, not by learning algorithms. To get you started, here's a step-by-step example of one way to insert a first layer corner.

Step 1 The blue/red/white corner is sitting in the bottom layer (the blue part is facing the bottom so we can't see it in this picture). Turn the blue face 90 degrees anti-clockwise.

Step 2 Now your cube should look like this. Move the D face 90 degrees anti-clockwise to line up the blue/white edge with the blue/white/red corner.

Step 3 Now that the blue/white edge and the blue/white/red corner have been lined up, re-form the white cross by turning the blue face 90 degrees clockwise.

Step 4 Now the blue/white/red corner is in its correct place.

Here are some tips for inserting the top layer corners:

• Start with a first layer corner that is sitting in the last layer.
• If there are multiple first layer corners in the last layer (there usually will be), start with one that does not have the white part of the corner on the face opposite the white face. Or, if you were using a different colour for the cross ('colour X'), start with a corner that does not have the 'colour X' part of the corner on the face opposite the 'colour X' face.
• When working with a first layer corner piece that is in the first layer (but in the wrong first layer corner position), you will need to get it out of the first layer into the last layer, then insert it into the correct first layer corner position. The same principle applies if a first layer corner piece is in the correct first layer corner position but needs to be flipped around. You need to get it out of the first layer (ie. into the last layer), and then re-insert it into the first layer the correct way around.

This is what the first layer should look like when finished.

The Middle Layer

The middle layer consists of one stage:

1. Insert the 4 middle layer edges (each edge is inserted individually).

You only need to learn one algorithm (plus the mirror algorithm) for the second layer. There are many more algs, but let's just learn the essential one first.
First, locate a middle layer edge that is currently sitting in the last layer. I'm going to use the blue/red edge for this example.

This blue edge cubie in the last layer is the blue/red edge cubie.

In this picture, U=white, L=red and F=blue. We can't see the other three faces, but obviously the R face is the one opposite the L face, the D face is opposite the U face and the B face is opposite the F face.
Now, position the blue/red edge piece so that the colour on the side of the cube (blue in this case) lines up with it's centre. Now perform the following algorithm: D L D' L' D' F' D F
If the blue/red edge piece was flipped the other way so that the blue was on the bottom rather than the red, you would position the cubie under the red centre and perform the following alg: D' F' D F D L D' L'. This is the mirror of the previous algorithm. The axis of symmetry lies diagonally across the white face, and along the line which divides the blue face and the red face.

What if the edge piece is not in the last layer? The instructions above assume that the middle layer edge piece you are inserting is sitting somewhere in the last layer. If some middle edges are in the last layer and some are in the middle layer in the wrong spot, always start working with the edge pieces that are in the last layer. After you've done this, sometimes (but not too often) you'll be left with a middle layer edge piece that's in the middle layer but in the wrong spot. In this situation, you can use the same middle layer algorithms from above (D L D' L' D' F' D F or D' F' D F D L D' L') to insert another edge piece into the middle layer edge position, thereby knocking the middle layer edge piece out of its spot and into the last layer. Once you've done this, the middle layer edge piece is in the last layer and you can deal with it in the usual way. There is a short-cut to this problem, but as this is a beginner solution with minimal memorisation, I haven't included it here. If you really want to learn it, take a look at Case Dd2 on Dan Harris' site.

The red/blue middle layer edge piece is in the middle layer but not oriented correctly. It needs to be moved to the last layer, then put back into the middle layer in the right way.

The Last Layer

The last layer ("LL") is done in 4 steps:

1. Orient the edges (2 algs) - i.e. form a cross on the D face.
2. Permute the corners (1 alg) - i.e. get the corners in the correct position in 3D space (don't worry if they still need to be rotated).
3. Orient the corners (1 alg + mirror alg) - i.e. flip the corners.
4. Permute the edges (1 alg) - i.e. swap the edges around. The cube should now be solved! :)

All last layer algorithms are performed with the cross (i.e. the first layer - white side in this example) on the bottom.

Orienting the LL Edges

Once you have completed the first two layers ("F2L"), hold the cube so that the white side is on the bottom. The white side will be on the bottom for the remainder of the solution. This means that the white side is the D side for all last layer algorithms.
On my cube, white is opposite yellow, therefore yellow is the U face for all last layer algorithms on my cube. Note that your cube may have a different colour opposite white (e.g. blue). Now have a look at your last layer, and in particular, look at the last layer face - there are 4 possible patterns of LL edges that you may see.

State 1

State 2

State 3

State 4

Unlike with the initial cross (where all the edges must match up with the white centre and with the centres on the middle layer), here all you need to worry about is getting all the last layer edges matching up with the last layer centre. It doesn't matter if the other colour on the LL edge piece does not match up with the colour on the middle layer centre. Also, ignore the LL corners too. It doesn't matter what they are doing at the moment. Now, let's consider each of these LL edge states separately.

State 1 All the edges are already oriented correctly. Move on to permuting the corners.

State 2 We are going to re-orient our faces for this algorithm. The face you are looking directly at in this picture is now the U face (it was the D face for when you were doing the second layer edges). Perform the following algorithm: F U R U' R' F'

State 3 As with State 2, the face you are looking directly at in this picture is now the U face. Perform the following algorithm: F R U R' U' F'

State 4 State 4 is really a combination of States 2 and 3, so all you need to do is perform the algorithm for either State 2 or State 3. Once you've done this, you'll see that your LL edges now look like State 2 or State 3, so just perform the appropriate algorithm and you will have a cross on the LL.

Permuting the LL Corners

The two possible states are:

• two adjacent LL corners need to be swapped; or
• two diagonal LL corners need to be swapped.

These are the only two possible states. If you cannot identify one of these two states with your LL corners then one or more of the following must be true:

• You have not finished the F2L.
• Someone has ripped out a corner of your cube and put it in the wrong way.
• Someone has ripped off some of your stickers and put them back in the wrong place.
• You are not looking hard enough. ;)

Swapping adjacent corners
Hold the cube with the white side on the bottom, and the two corners to be swapped are in the front right top and the back right top positions. Perform the following algorithm: L U' R' U L' U' R U2. To see an animated version of this algorithm, see the first algorithm on Lars Petrus' Step 5 page. On Lars' site, the algorithm is being executed from a slightly different angle (the two corners being swapped are front-top-right and front-top-left), but it is the same exact algorithm.
Swapping diagonal corners
Swapping diagonal corners can be done by executing the adjacent corner swap algorithm twice. Perform it once to swap any two LL corners. Re-examine you cube and you'll see that now there are just two LL corners that need to be swapped. Position it correctly for the final LL adjacent corner swap and perform the LL adjacent corner swap algorithm.

Orienting the LL Corners

There are 8 possible orientation states for the LL corners. One is where all 4 corners are correctly oriented. The other 7 look like this.

State 1

State 2

State 3

State 4

State 5

State 6

State 7

State 1. Twisting three corners anti-clockwise R' U' R U' R' U2 R U2

State 2. Twisting three corners clockwise R U R' U R U2 R' U2 To see an animated version of this algorithm, look at Lars Petrus' Sune algorithm.

States 3-7
Once you know the algorithms for States 1 and 2, you can solve any LL orientation State. The remaining States can be oriented using a maximum of 2 algorithms. You will need to do one of the following (i) the State 1 algorithm twice, (ii) the State 2 algorithm twice, (iii) the State 1 algorithm, then the State 2 algorithm, or (iv) the State 2 algorithm, then the State 1 algorithm.
In a previous edition of this solution, I had said that I'm not going to tell you exactly how to combine the State 1 and State 2 algorithms to solve States 3-7. My reason for this was because it is important that you try to understand how the State 1 and the State 2 algorithms work, and that once you do understand them you will be able to work out how to use them to solve all the States. I still believe this, however, I received emails from a few people who were having trouble with States 3-7, so I decided to write some extra tips. I still suggest that you try to work out States 3-7 by yourself, but if you are really stuck, have a look here: Orienting the Last Layer Corners: further tips.

Permuting the LL Edges

There are 5 possible permutation states for the LL edges. One is where all 4 edges are correctly permuted. The other 4 look like this.

State 1 R2 U F B' R2 F' B U R2

State 2 R2 U' F B' R2 F' B U' R2 This is almost identical to the algorithm for State 1. Only difference is the 2nd move and the 2nd last move. For an animated version of this algorithm, see the Lars Petrus' Allen algorithm. The algorithm is being executed from a slightly different angle, but it is the same exact algorithm.

State 3 Apply the algorithm for either State 1 or State 2. Re-examine your cube and it will now look like State 1 or State 2.

State 4 Apply the algorithm for either State 1 or State 2. Re-examine your cube and it will now look like State 1 or State 2.

And that's all you really need to know to solve the Rubik's Cube!