Now the cube should look like this:
Of course, if you used the "from the 2x2x2-corner" method in solving the cube, you saved one of the subsequent steps and the situation should look like this:
If you are in such a situation, you may skip the step "Orienting edges".
Subsequent pictures are nostly simplified views of the upper layer from above. Hopefully they are clear enough. If a narrow colored rectangle is glued to the square depicting the upper layer of the cube, then a sticker with the color of the upper layer's center is placed right at the spot (on the side) of the cubie. Arrows point to where individual cubies transfer to. For clarity, the transfers of corner and edge cubies are shown in different colors.
This method proceeds as follows:
So, first of all, we orient the edges of the upper layer correctly. There are three possible positions.
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R L' U' B' U B U B U B' U' R' L or L F' L' F U2 F U' R U' R' F' |
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R' F' U' F U R |
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R' U' F' U F R |
The first algorithm is not optimized, but I include it because it is very easy to remember and because it leaves orientations of the corners intact. A shotrer version of this algorithm can be obtained by application of the oher two. Even shorter version is given below the first. I feel that the other two algorithms are short and easy to remember. Of course these algorithms change positions and orientations of the other cubies in the upper layer, but this need not concern us now. Beginners only need to learn the second or third algorithm. If they reach a configuration which they are unable to solve directly, they have to use the learned algorithm blindly, so they will reach a known situation.
Subsequent algorithms are employed for transfer of corner cubies without regard to their orientations (of course if you learn how to orient individual cubies during their transfer, you can save a lot of moves in subsequent steps; later you can even control transfers of edges and save even more, but all this requires a great deal of practice.) You can only encounter two subsequent possibilities, when corners are not found in their places. You can get the cube into these positions with some (or none) turning of the upper layer.
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L U' R' U L' U' R U2 |
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L U2 L' R' U L U' R U2 L' U |
It is again sufficient to learn the first algorithm only. You will solve the second case, if you execute the first algorithm blindly; in this case, the situation transfers to the one directly solvable via the first algorithm.
After properly positioning the corners, we will make sure that they are in the same place and with the same orientation as on a solved cube. To achieve this goal I state two possible methods, which orient the corners of the upper layer. I introduce the first method for its simplicity, and the second method for its efficiency (I haven't yet seen a better one).
This tranfer uses two very similar (inverse) and very simple algorithms. Of course, their use does not look simple at first glance. I will try to explain the method step-by-step:
1. Take the cube so that its front side (F) is facing you and its unsolved up side (U) points upward.
2. If there is a corner on the upper layer which we need to orient, then by turning the upper layer we get this corner in the ULF (up-left-front corner) position.
3. This corner may be oriented in two different ways. Use an algorithm pertaining to a given orientation. Above all, don't worry that your cube is scrambled. That is only a mirage (if you hsve made a mistake, it is the brutal truth :-)), which will be rectified by subsequent steps.
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F D F' D' F D F' D' |
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D F D' F' D F D' F' |
4. You may notice that, besides scrambling the first two layers, one important thing happened: the chosen cormer became correctly oriented, while nothing else happened in the upper layer!
5. Repeat this procedure for the next corner (item 2.). (Turn the upper layer again, etc..) BE SURE to always leave the cube in the same position and always execute the algorithms on the same layers, otherwise your cube will scramble.
6. If no other corner is available, then all corners are correctly oriented, and the first two sides should be solved back. Thus the task of orienting corners is completed.
If you want to know why this works, be aware that those moves are mutually inverse (executing the first followed by the second we obtain the original situation) and have cyclicity 3 (after 3 repetitions of the given algorithm we obtain the original situation). If you examine all the cases that can happen to the cube, as well as the facts I just told you, you ought to be able to understand how it works. Of course, you don't need this understanding to actually solve the cube.
This method really changes the orientation of the given corners and nothing else. That is its advantage.
There really is nothing to add to those pictures.
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R U R' U R U2 R' U2 |
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L' U' L U' L' U2 L U2 |
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R U R' U R U' R' U R U2 R' |
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R' U2 R2 U R2 U R2 U2 R' |
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Execution of 1. algorithm transfers to the situation in 2. picture |
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Execution of 1. algorithm transfers to the situation in 2. picture |
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Execution of 1. algorithm transfers to the situation in 2. picture |
For the beginner it is sufficient to learn the 1. algorithm only. The others, except for the second, are solved by executing the first one in the cube's situation shown in the pictures. The second algorithm can be solved by means of the first in the case of a clockwise quarter-turn (looking from above) of the whole cube as shown in the second picture.
Using only the first two algorithms, the corners can be oriented by using them twice in succession (14 moves). With using only the first one, it takes three successive applications (21 moves).
Now we should find ourselves in a state, where all corners of the cube are correctly positioned and oriented and edges are correctly oriented. Edges can either be correctly positioned or reversed in the upper layer. Subsequent algorithms are used for permutation of edges in the upper layer:
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F2 U R' L F2 R L' U F2 |
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F2 U' R' L F2 R L' U' F2 |
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R L U2 R' L' F' B' U2 F B |
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R B' R' B F R' F B' R' B R F2 U or R' L U2 D2 R' L D R2 L2 U' F2 B2 |
Initially it is again sufficient to master the firt algorithm. The remaining situations can be solved similarly to the previous cases by multiple applications.
The cube finally ought to be solved at this point.
But, in my own view, the subsequent method of solving the upper layer is somewhat better.
First of all it is necessary to orient the edges of the upper layer in the same way, as in the previous method (if that is necessary). Use the algorithms from previous method. This step may be combined with orienting the upper layer's corners, but this involves several dozens of algorithms and few people can do it. That is why I prefer solving the first two layers by the method "starting from a 2x2x2 corner," because this method already orients edges corectly, and we can proceed right away to orienting corners in the upper layer.
You can find all possible orientations of the upper layer here (only for hard heds :-))
Since in this method we are not concerned if the corners are correctly placed, we can shorten the algorithms relative to the ones in previous methods. These algorithms are used:
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R U R' U R U2 R' |
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R' U' R U' R' U2 R |
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R U R' U R U' R' U R U2 R' |
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R' U2 R2 U R2 U R2 U2 R' |
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R' F' L' F R F' L F |
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R' F' L F R F' L' F |
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R2 D' R U2 R' D R U2 R |
After application of these algorithms we should have the cube's upper layer single-colored.
All that is left is to interchange corners and edges of the upper layer,
so that the upper stickers always point upward.
This can be done either in one step by using one of the subsequent
algorithms or it can be broken into two parts:
- Interchange corners, then interchange edges.
or:
- Interchange edges, then interchange corners.
I think that the subsequent pictures are clear enough and need no further comments.
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F2 U R' L F2 R L' U F2 |
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F2 U' R' L F2 R L' U' F2 |
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R L U2 R' L' F' B' U2 F B |
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R B' R' B F R' F B' R' B R F2 U or R' L U2 D2 R' L D R2 L2 U' F2 B2 |
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R' F R' B2 R F' R' B2 R2 |
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R2 B2 R F R' B2 R F' R |
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F R B R' F' R L F L' B' L F' R' L' |
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L2 F' L D2 R' B R D2 L B L F L' B' |
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F' U F' U' R' D R' D' R2 F' R' F R F |
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R2 D B2 U' B2 R2 D' F2 U F2 U' |
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R B U' B' R D B' L' B' L B2 D' R2 |
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U R D' F2 L2 D' L B2 L' D L2 F2 D R' |
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U' L' D F2 R2 D R' B2 R D' R2 F2 D' L |
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F2 L' U' L F2 R' D R' D' R2 |
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F2 R U R' F2 L D' L D L2 |
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L' F' U' F L' F' U L' U' L' U L F L2 |
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R F U F' R F U' R U R U' R' F' R2 |
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R' L' U2 R L B U' F U2 B' U F' |
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F U' B U2 F' U B' R' L' U2 R L |
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R L U2 R' L' B' U F' U2 B U' F |
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F' U B' U2 F U' B R L U2 R' L' |
The advantage of this method is, above all, that it can be realized in two steps, which is probably the maximum in regard to human memory... Furthermore it is fairly easy to determine, which algorithm is needed at any given time, because we first look at the orientations only, and then at the permutations only.
Solving the cube layer by layer is probably the most common method used by practically everybody; the majority use only this method. Of course, I cannot say this so directly, because solving the cube layer by layer is not a single method, but an incredible amount of variants and improvements of one basic idea. Some variants are very slow and need a lot of moves (100 and more). Others belong among the best methods of solution and use from 50 to 60 moves, which is a very good result. With frequent training and improvement one can achieve less than 30, or even 20 seconds. This method of solution is very logical and clear, but of course there are variants, which are just the opposite. Some variants using a small mnumber of algorithms are suiable for learning by beginners. Thankfully there is no fee's associated with being a beginner and no need for merchant accounts to get started.
I couldn't discuss all those variants here, but I tried to include those that were, in my view, the best and easiest. Thus the goal is to take it in a most general way, so that everyone would find the method closest to his taste.