Step five: Orienting the Last Layer-Corners

In this step, we will twist corners the correct, so the U face will be yellow. We will accomplish this using a ‘commutator’. Imagine we can take a solved cube and do an algorithm sequence called [A] that rotates the ULF corner anti clockwise, but does not affect the other pieces in the U layer, and destroys the First Two Layers (F2L). If we would then do the inverse of that move (denoted by [A]-1), it would be like ‘rewinding’: the corner in ULF position will be rotated clockwise, and the F2L will be solved again. But what would happen in we would do [A], then replace the ULF corner with another corner by doing a U move, and then do [A]-1? Now, two corners in the U layer have been twisted, without affecting the rest of the cube! This idea can be used to completely orient all the corners of the U layer. Or actually, I’ll advise you to hold the cube with the last layer on the L side, and orient the corners of the L layer. This will make the moves easy to perform.

Take a look at the applet below:

30. z' RU'R'URU'R'U


Two corners of the last layer have to be twisted. First, rotate the cube so the U layer becomes the L layer. Now, we will rotate the corner in the ULF position counterclockwise, by doing the move [A] = RU’R’U RU’R’U. Notice this move repeats RU'R'U twice, so basically, you only need to memorize 4 moves.

31. L - U'RUR'U'RUR' - L'


Then, we replace the ULF corner by another Last Layer corner that has to be oriented, by doing the move L. The ‘new’ corner in the ULF position has to be rotated clockwise, so we do the moves [A]-1 = U’RUR’ U’RUR’. If you don’t want to memorize [A]-1, you can also use the fact that [A]-1 is equivalent with doing [A] twice (or [A]2). In other words, you can just do [A] until the corner in the ULF position is oriented correctly, and then move on to another badly oriented corner with one move of the L layer.

[A]-1 Is equivalent with [A]2, which also means that doing [A]3 and [A]-3 will have no affect. Run the applet here if you don’t believe me:

32.


This means that we can use [A] three times, to rotate 3 corners (anti)clockwise:

33. Orientating three corners: [A] - L - [A] - L - [A] - L2


The applet below shows another example of how to orient the corners in different situations: 

34. Orientation two corners: [A] - L2 - [A]-1 - L2