4.1 Solving the first two layers
4.1.1 Solving the cube layer-by-layer
In majority of cases the layered method is the method one encounters, if one decides to master the solution of the Rubik's cube. The probable reason is, that one can learn the method in stages: first layer - second layer - third layer. Moreover the beginner can easily monitor his progress. The difficulty increases with each successive layer, so it is easy to start.
4.1.1.1 Solving the first layer
In most solution manuals, the first step toward solving the first layer is correct placement of the edges in the upper layer, as shown in the picture.
It is important to realize, that it is not sufficient to have the white stickers point upward. We also have to be sure that the remaining colors of these edges are identical to the colors of the centers of the sides adjacent to the upper layer. It is not necessary to remember special methods in order to reach this stage. Just do a little thinking, but to make it clear, I will show you how to reach this stage via an example.
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| Fig. 1 | Fig. 2 | Fig. 3 |
If the edge, which you want to turn into its position in the upper layer, is in the lower layer, then the situation can be transfered to any one of the situations shown in the first two pictures by turning the lower layer. If one has the situation shown in Fig. 2, then, in order to place the edge in the correct position, it is sufficient to rotate the front layer by a half-turn (F2). If one has a situation shown in Fig. 1,then one executes this sequence: D R F' R'. Try to figure out for yourselves, how this works. Later on, you can save some moves by judicious thinking. Should the edge be found in the middle layer (as, for example, in Fig. 3), you can either move it to the lower layer with a sequence B' D B, or, for this particular case, it is better to use B' D2 B; from the lower layer you should know how to put it in correct position. Continue doing this, until all edges are in their correct positions. Should an edge be in a wrong position, you can place it in correct position by, e. g., placing the said edge in the lower layer (e. g., for the UF edge, execute F2) and by doing so, it transfers to the already-known case.
Next step is to place corners of the upper layer in correct position. The situation that can occur is depicted in the following picture.
This step may best be explained by looking at pictures, which probably show most cases that can occur.
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| R' D' R or D F D' F' | R' D R, then D2 R' D' R | R' D' R, then D R' D' R | F D F', then D' F D F' |
In the last two pictures you can see how situation with mirror-symmetry are created; just mirror-reflect them. Should you by any chance have a corner already in the upper layer, but positioned incorectly, then you have to get it in the lower layer (similar to pictures 3 and 4), and from there you can place it correctly.
After some time you will surely find more efficient versions of the algorithms for each special case that will arise. By now you should have completed the upper layer of the cube. If not, keep trying and thinking, the final result will surely materialize.
4.1.1.2 Solving the second layer
Now we turn the whole cube so that the solved layer points downward, so we can work with it better. We don't have to show the lower layer (white in our case) because it will no longer change.
Now we will try to place the 4 edges in the middle layer in order to complete it. We have to place the edges with correct orientation. The following picture shows placing the edge, which we want to move to correct position (keep turning the upper layer until you reach these or mirror-reversed situations - it may not always work, see below).
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| move 1: F2 U2 F U2 F2 |
| move 2: U R U' R' U' F' U F |
During execution of the first sequence the edge, colored dark-grey in the picture, is not preserved. This sequence, however, is shorter, so it is a good idea to apply it to the first 3 edges, and use the second move, which preserves all edges in the middle layer, for the last edge. If you don't have any edge from he middle layer in the upper layer, which you could place, this indicates that all edges are in the middle layer. Of course, such edges may not be in correct places, and if they are, their orientations may be wrong. If such a situation occurs, these wrongly-placed edges need to be transfered to the upper layer by applying move 1 and 2, when we put the edge of the upper layer in place instead of the wrongly-placed edge of the middle layer. Thus the two edges get interchanged and we can place the "pushed-out" edge in correct position. It is evident that, if a mirror-reversed situation occurs, we have to execute mirror-reversal operations.
In this manner we should obtain a cube with two layers solved. We can learn other (better) ways or we can proceed to learn solving the last layer.
4.1.2 Solving with the aid of a working corner
This method was developed in order to reduce the number of moves needed to solve the second layer. The basis is constructing a cross in the upper layer, that is, placing the edges of the upper layer in correct positions, which we already know from some of the previous chapters. This is followed by placing three corners in the upper layer, which we also know from some of the previous chapters. Should you place all four corners, nohing would happen, but in the subsequent step you would destroy on corner, chosen as your working corner. Now we again turn the cube upside down. The situation should look like this:
With the aid of such a working corner you can place 3 edges in the middle layer. Always put the working corner under the edge in the middle layer, at a spot where you want to place correctly positioned and oriented edge from the upper layer. You will achieve this by turning the lower layer. The following scheme shows, how to put the edge from the upper layer into correct position in the middle layer.
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| R U R' |
The situation may of course be mirror-reversed, in which case we use the mirror-reversal moves. If we don't have any edges from the middle layer in the upper layer, then all such edges must be in the middle layer, but don't have to be correctly positioned or oriented. In this case we place the working corner under wrongly positioned or oriented edge and again execute, e. g. the R U F' move. This operation will transfer the edge into the upper layer and from there we can place it in the middle layer. In this step we recommend to place the 3 edges not under the working corner. In other words, if you put given 3 edges of the middle layert in correct place, and turn the lower layer such that the colors of the centers of the side faces are identical to the colors of the stickers belonging to cubies in the lower layer, the unsolved working corner together with the edge should be above it. The subsequent picture shows the situation:
To fill the working corner and the edge above it we use the method described in previous chapters (sequences for putting the corner in the upper layer do not change edges in the middle layer, except for the edge adjacent to the corner). This first two layer should again be completed. We can either master other methods or proceed to solving the last layer.
4.1.3 Upper edges, then upper corners together with side edges
The basis of this method is again solving the upper cross, that is, placing the edges on the upper layer in correct positions. Instead of putting the corners themselves into the upper layer we employ a whole series of algorithms, which place the corner and its adjacent edge in their positions simultaneously. Most of these algorithms are relatively straightforward, and are either discussed in previous chapters or derived from them. Of course, it is possible to invent other algorihms, that can be quite useful and efficient. I will present a few examples here, you can figure out others after some time.
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| R2 D2 R D R' D R D2 R | R2 D2 F' R2 F D2 R D' R | R' D R' D' B' D B R2 |
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| D R' D R D2 R' D R | R2 D' R2 D' R2 D2 R2 | D R' D' R D' F D F' | R' D R F D2 F' |
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| F D' F2 R F R' | R' D' R D R' D' R |
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| F' R F R' | R' D R D2 F D F' | D' F D2 F' D R' D' R |
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| D2 R2 D2 R D R' D R2 | D R' D R D' R' D' R | F D2 F' D' F D F' |
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| D' F D2 F' D2 F D' F' | D' F D' F' D R' D' R | R' D' R |
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| D2 R' D' R2 F' R' F | D' F D F' D2 F D' F' | D F D2 F2 R F R' |
Now only those situations can occur, where a corner or an edge we want to work with are in upper or middle layer; then we will either choose a different pair, or we will remove them from these positions by some method mentioned in previous chapters. All situations can have their mirror-reversed doubles. Then we use algorithms for mirror-reversal. Now we can proceed to solvin the last layer.
4.1.4 Beginning with a 2x2x2 corner
This method differs quite a bit from those discussed previously. It does not progress from some layer to one adjacent to it. Instead one constructs a small and continuous part of the cube - a corner consisting of 2x2x2 cubies (1 corner cubie, 3 edge cubies and 3 adjoining center cubies). After this step we can freely rotate three mutually orthogonal unsolved layers and solve the cube gradually. We try, if possible, not to rotate any other layers, since doing so may scramble the cube.
In the next step, by rotating the three free layers, we increase the 2x2x2 corner to a 2x2x3 block by adding another corner, two edges and one adjoining center. After this step we have two unsolved, mutually orthogonal layers. By turning these two layers the subsequent goal is to increase the 2x2x3 block to a 2x3x3 block, or complete the first two layers. Needless to say, some of the edges in this step may have wrong orientation, so that they cannot be placed and oriented properly. We have to reverse such edges before completing the step. We can then finish the two layers by rotating the last two free layers.
After this step, the two layers are solved and we may proceed to solve the last layer. Since we have in a previous step reversed the edges, and rotating two layers does not reverse them, the edges on the last layer are correctly oriented, and this reduces considerably the number of algorithms needed to solve the last layer.
Let us proceed to actual execution of individual steps.
4.1.4.1 Solving the 2x2x2 corner
The goal of this step is to produce the situation on the cube, shown in the following picture.
This step can be performed "all together"(you will learn this with time) if you can divide it into the following smaller steps:
Example:
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1. (B U) (D) (B2) (F' R2) 2. (B U) (D R2) (L2 F2) (R2 F' R2) |
In this case parentheses represent previous parts of a method. In the first variant the method is not strictly followed, so that one may benefit from this. In the second variant the method is strictly adhered to.
At first glance the step does not look too easy, but it can be learned very quickly. Usually one just needs to think a little. With time one can find new and shorter combinations of the 2x2x2 corner, and therefore one should spend a lot of time thinking about improving and shortening the solution of this part....
In this step we can select an arbitrary corner and thus we have 8 possibilities (in previous methods there were 6 sides, and therefore 6 possibilities).
4.1.4.2 Solving the 2x2x3 corner
Our goal in this step is to get the cube in the state shown in the following picture.
Again this cstep is quite intuitive. We can use a method similar to the previous step. The matter is complicated by the fact, that we can only rotate three layers freely. With time you can find some handy algorithms, which temporarily disturb the 2x2x2 block. Initially, however, you should leave the 2x2x2 block undisturbed.
Again we can divide this step into these smaller steps:
Example:
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1. (F R' U) (F U2) (F U F' R' F2) 2. (F R' U) (F U2 F2) (U F2 R' F2) |
The parentheses again mean individal steps of a previous method. The first variant does not adhere to the method, while the second variant does. In this case we do not gain anything by not adhering to the method, but we save a few moves.
In this step we can choose the way to increase the 2x2x2 block to the 2x2x3 block "only" from three possibilities. Notice, that only one method offers the possibility of deciding and choosing the sides to continue with, even after first steps.
Remark: Some people find it convenient to sart solving the block 2x2x3 directly, but they save fairly little. Initially they have 12 possibilities for starting (12 edges of the cube).
4.1.4.3 Orienting remaining edges
Now the 2x2x3 block is solved, and we can work with two mutually orthogonal layers, which are unsolved. Now our goal will probably be to orient the edges, so that we may complete the two layers (the last one is still unsolved) by turning only two unsolved layers. We recognize the wrong orientation of edges by placing them in correct positions via turning only the two unsolved layers; the edges will have wrong orientation. The number of such edges is always even!
Finding such edges is a very slow process, so we will use another way to locate these edges:
If we label the colors of centers belonging to two unsolved layers A and B, then the wrongly oriented edge is an edge which fulfills at least one of these conditions:
There can be 2, 4 or 6 such edges. The subsequent pictures show how to place such edges correctly. Wrongly-oriented edges are shown in color, the already-finished 2x2x3 block is shown dark. Those are not all possible positions; these are positions with minimal number of moves for a given number of edges. We can obtain other siuations by turning one of the unsolved layers. There is only one situation that requires turning both unsolved layers, but you can figure out for yourselves what that is. Always try to get the incorrectly-oriented edges into some of the positions below and apply the pertinent algorithm.
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| L F L' | L F' L' | L F2 L' | L R' F L' R | L F' L2 U L |
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| L F2 L2 U L | L F L2 F2 L | L F L2 U' L | L' U L2 F L' | L R' F L2 R U' L |
Memorizing these algorithms is not necessary; just try to understasnd how they work. Initially it is sufficient to reverse the pair of edges by using the first (or second...) algorithm.
Notice, that only until now were we forced, for a while, to use the method destroying what we have done before in solving the cube.
4.1.4.4 Finishing the two layers
The goal of this step is to get the cube into the following state.
We have just two possibilities to choose from, which layer to continue with and which one to leave unsolved. We turn only two unsolved layers. We are not concerned about cubies whose stickers' color is the same as the color of their layer (we don't solve layers now). We keep the layer until the next step. We can again divide the method into subsequent smaller steps: (all edges and corners we are now talking about, do not have their stickers the color of the unsolved layer)
Example:
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(U2 R' U') () (R U' R2 U R) |
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(U' R U2) (R' U R U' R') |
Initially we have to do quite a lot of thinking about these steps, but they become automatic with time....
Now we have two layers solved, and we can proceed to solving the last layer.