Improving the Easy Cube Solving Method

Now that we know The Easiest Method to Solve the Rubik's Cube, it is time for the first improvements to it. But you should keep in mind that none of these improvements are sustainable. You will later learn better algorithms that achieve the same with less effort. So feel free to skip this article if you are sure that you want to learn the "real thing".

Improving the Final Edges Permutation

The last step of our easy method is the positioning of the edges of the last layer. We have done it like this:

This is by far the longest sequence of moves of our method. It consists of 48 individual turns. There must be an easier way.

Inside the Sexy Move

Let's look in more detail at the Sexy Move. We know that repeating it 6 times leads back to the starting position:

By intuition, we also know another fact: The effect of each single turn of the cube can be undone by turning the same layer in the opposite direction. And what works for one single turn, will also work for every sequence of turns. You can undo it by the reverse sequence of turns. So, how does the reverse Sexy Move look like?

We will refer to the sequence U R U' R' as RSexy' (RSexy prime) and to U' L' U L as LSexy' (LSexy prime) from now on.

It should be clear by now that doing the sequence RSexy RSexy' or LSexy LSexy' will simply return to the initial position:

The same must be true, when we just swap the order of moves and do the prime version first, with RSexy' RSexy or LSexy' LSexy

That means that after RSexy', you have to perform an RSexy to get back to the original position.

Now remember that after six repetitions of RSexy you also get back to the original position. And after five repetitions of RSexy you have to perform one more RSexy to get back to the original position. And that means: Doing five times RSexy is the same as doing RSexy' just once!

Optimisation With RSexy' and LSexy'

When we apply that to the edge permutation problem, we now understand that we can simply replace the five repetitions of RSexy and LSexy with just one single RSexy' respectively LSexy'. That saves us 40(!) turns.

Let's look at that in action:

Controlling the Permutation Direction

Our original strategy was that we have to do the recipe for the final move of the edges either once or twice.

As it turns out, that not only RSexy LSexy RSexy' LSexy' cycles three edges but also the mirrored sequence LSexy RSexy LSexy' RSexy'. See the two variants:

With both variants, two edges travel to the next side diagonally. One edge travels all across the cube to the other side. You start the move on the side where this piece travels.

In other words: If one edge has to travel all over the cube to the right side, you start with RSexy. If one edge has to travel to the left side, you start with LSexy. That again saves 8 turn by average.

Improving the Corner Orientation

We orient the corners of the last layer with the cube upside down, and performing the Sexy Move in only one slot over and over again. But inbetween, whenever a corner has been oriented correctly, we rotate the bottom layer and move the next incorrectly oriented corner into that slot.

Choosing the Best Sexy Move

Sometimes a corner is correctly oriented after just two Sexy Moves. See this example with three twisted corners:

But there is also an unlucky case where you have to perform four Sexy Moves for each corner:

Compare the initial position of the two cases. In the upper, lucky case, the stickers in the colour of the bottom faces to the right. In the lower, unlucky case, it faces to the front. But what if you use LSexy instead of RSexy but still in the same slot?

The rule of thumb is: Rotate the cube so that the sticker in the bottom colour faces to the side, not the front and perform the Sexy Move on that side.

Choosing the Right Direction Sexy Move

But the last optimisation is not useful, when not all corners have to be rotated in the same direction. Most of the cases are mixed cases, where the corners have to be rotated to different directions.

When a corner has to be rotated into the "wrong" direction, we have to perform the Sexy Move four times. But we now know already that performing the Sexy Move four time is equivalent to performing the reverse Sexy Move RSexy' respectively LSexy' twice.

That leads us to the final optimisation. If the sticker in the bottom colour faces to the side, perform the normal Sexy Move twice. If it faces the front, perform the reverse Sexy Move twice. Whether you use RSexy/RSexy' or LSexy/LSexy' is up to you. Let's look at both variants:

Conclusion

These are the obvious improvements to our solving method. They come at the price of learning another algorithm, but an easy one, because it is the one that we have alreay learned, backwards.

Although it is by far not an optimal method, with some practice and a decent cube, you should be able to solve it in 30-40 seconds as a personal best.