The first step is understand the notation and Figure 1 shows the four terms for the four faces that will be used in the solution (D for Down and B for Back are not used):
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| Figure 1 |
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| Figure 2 |
STEP 1: the first step is to create what is called the Daisy. See Figure 2. This is quite straight forward and doesn't require any special algorithms as will be needed later. It can be completed simply by careful inspection.
We then rotate the bottom two layers so that one of the centre square colours matches that of a white edge. Once this has been done, the face is then rotated through 180°.
This is repeated for the other three faces. Figure 3 shows an alignment of the green centre square with the corresponding green/while edge. The face is now ready for the 180° twist. This completes the first step where we have a white cross.
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| Figure 3 |
STEP 2: the idea in this step is to complete the white face. For this we use what is called the RIGHTY ALG where ALG stands for ALGORITHM. This algorithm involves a right face clockwise turn, an upper face clockwise turn, an anticlockwise right face turn and an anticlockwise upper face turn. This is depicted as R U R' U'.
The white cross is placed face down and the corner whites are first positioned above where they need to be and the righty alg is applied repeatedly until the white corner pieces are in their correct positions. At the end of this step, the while face is now complete.
STEP 3: in this step we complete the bottom two layers. Additionally, this will involve the use of the LEFTY ALG which is characterised by L' U' L U or, in other words, a left face anticlockwise turn, an upper face anticlockwise turn, a left face clockwise turn and an upper face clockwise turn. In addition to the use of the lefty and righty alg, the cube will need to be spun clockwise or anticlockwise. See Figure 4:
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| Figure 4 |
To move an edge to the right, the steps shown in Figure 5 must be followed where R stands for the righty alg and L stands for the lefty alg:
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| Figure 5 |
To move an edge to the left, the steps shown in Figure 6 must be followed:
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| Figure 6 |
At the end of this third step, the bottom two layers should be complete.
STEP 4: this step involves creating a yellow cross, assuming one has not already been created accidentally. There are two possible scenarios that need to dealt with. The first occurs if one arm of the cross is already complete. See Figure 7 for the algorithm to complete the cross in this scenario (note that the already completed arm needs to be held horizontally as shown):
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| Figure 7 |
If neither of the two arms of the cross are complete and there is more than the central yellow square uppermost, then the algorithm show in Figure 8 can be used (note the empty corner must be in the bottom right position and care needs to be taken with lower case f which means the front two faces are rotated not just the first face):
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| Figure 8 |
Sometimes only the central yellow square is uppermost. In this case, complete the algorithm for the line: F R F'. This will produce the corner and the f R f' can then be applied.
STEP 5: this step involves getting the corner pieces of the last layer into their proper positions. Sometimes, two adjacent corners will need to be interchanged. In this case, use the algorithm described in Figure 9 (note that the corners to be swapped must be positioned on the right):
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| Figure 9 |
For a diagonal swap, the previous step needs to be completed once in any orientation (provided yellow centre is uppermost) to be produce a situation where an adjacent swap is required. Position the corners to be swapped on the right and repeat the algorithm again.
STEP 6: now that the corner pieces are correctly positioned (but in some cases twisted), we want to complete the yellow face. The first step is to flip the cube so that the white face in uppermost. See Figure 10.
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| Figure 10 |
Next, identify a corner that needs to be twisted and place it on the bottom right. See Figure 11.
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| Figure 11 |
Finally, carry out the RIGHTY ALG until the corner is correctly positioned. See Figure 12:
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| Figure 12 |
After doing this for one corner, rotate the bottom layer so that another yellow sticker that needs twisting is in the bottom right position. Keep doing this until the yellow face is completed.
STEP 7: now the edges on the last layer need to be positioned correctly. There are two possible cases to deal with. The first is when one edge is positioned correctly and the second is when no positions are positioned correctly. The first case algorithm is shown in Figure 13 (sometimes this needs to be done twice).
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| Figure 13 |
For the case where no edges are correctly positioned, carry out the above algorithm to get an edge that is correctly positioned and then repeat the algorithm with the correct edge facing forward. And that's it. These moves should lead to a completed cube.
For a mathematical examination of the Rubik's cube, the following Numberphile video is recommended viewing:
