Given that you can solve the top layer, there is a really nice hack that lets you work out the rest of the puzzle. Say your top layer is completely solved, exc

Given that you can solve the top layer, there is a really nice hack that lets you work out the rest of the puzzle. Say your top layer is completely solved, except that one of the top side pieces is flipped 180 degrees. You know a sequence of moves that will fix that piece, to solve the top layer. If you do that sequence and then run the same sequence backwards, you will get the cube back to the same place, first with that side piece fixed, and then rotated 180 degrees again. But here is the trick: Before you run the sequence backwards, rotate the top! You will restore the cube exactly to where it was, except that now TWO of the top side pieces will have been flipped 180 degrees! This is referred to as a commutator operation, and is what Ryan Heise explains. Variants of this trick will let you rotate two corners in opposite directions, and re-position two top pieces and one bottom piece. One last thing: when solving the bottom layer, you may find yourself unable to re-position cubies into place using 3-way moves. Just rotate the top layer 90 degrees and you will then be able to proceed to completion.