The usual solution to Rubik’s cube is an algorithm – do this, do that and you have the first layer, then second layer, and last layer is made via set of arbitrary moves.
I always had problem remembering those, but when the sexy move (play animation here:
R U R' U' ↗) was introduced to me I thought of an easy-to-remember solution.
If you want to come up with it yourself, then I have one simple advice: analyze what sexy move does, including when it is repeated several times; that should be almost enough to solve the cube.
The sexy move analysis
The move consists of
R U R' U' which rotates only right and top side.
(You may also perform its mirror variant
L' U' L U.)
Doing this move once does too many things to explain easily, but we notice that it only affects cubes along the front-right, top-right, and the top-back edge of the cube.
The constituent parts of the move may be easily described by what happens when the move is repeated as follows.
- 6 repetitions ↗ – returns back to its original state
- 3 repetitions ↗ – returns edge-cubes to original states, swaps corners of the front-right and top-back
- 2 repetitions ↗ – corners just rotate, the three edge-cubes are permuted (in a cyclic manner)
- 1 repetition ↗ then cyclically rotates edge-cubes and swaps corners of the front-right and top-back, while rotating them
First solve all edge-cubes, then do all of the corners.
The two rows can be done by anyone (leave corners unsolved), leaving only the top side unsolved.
The edge-cubes of the top side may be manipulated by rotating on side so that the sexy move can be applied (e.g.
F [R U R' U'] F') – applying this we can rotate the top side’s edge-cubes so that the same color is on top.
Due to how the cubes move in the sexy move, try doing
[R U R' U'] T' [R U R' U'] T T (borrowing an edge from the solved part); having this its simple to solve edge-cubes completely.
Now we shall use only 3x sexy move (which preserves edge-cubes), and manupulate the corners so that they are swapped as we want. First, move all white corners to the white side (easy). Next, within that side, move orange-white pieces to the orange-white side (medium). Last, swap corners within the last unsolved part to their correct place and use the solved part to swap only two corners there and back again; parity ensures that this will finish smoothly.
Having all cubes in their good places, we use a trick presented by Mathologer ↗ using the sexy move – the cube at bottom-fron-right corner is the only cube affected by the move, so use the move twice to rotate it; shift the bottom side to affect another cube, and do the move in reverse. This fixes all cubes except of the bottom ones, which remain rotated; this was our goal. Rotate the cube so that the bottom part contains two badly rotated pieces, and do this to fix them.
That’s the solution. Using one move while just rotating some faces to participate in it, and then returning them back.