Putting the corners in place

The last four pieces

Everything is solved except the four top corners, and they're wrong in two ways at once: they sit in the wrong spots, and their yellow points wherever it pleases. So we'll split the job in two. This chapter puts the corners in their spots; the next one twists them.

The top edges are already where they belong, so don't turn the top on its own in this step: every stray U knocks them out of line. Use nothing but the algorithm, and bring the corner you need to the front by turning the whole cube.

How to tell a corner is home

A corner is in its place when its three colors match the three centers around it. Which way those colors face, and how the corner is twisted, doesn't matter yet.

Take the top right front corner. Its colors are yellow, green and orange, and the centers around it are yellow, green and orange. They match, so the corner is home, even though its yellow sticker faces forward. The other three corners here, though, are out of place. Check each one.

Go around in order so you don't skip any: front right, back right, back left, front left. For each one, name the corner's three colors to yourself, then the three centers. It's slow at first; after a couple of solves you'll see it at a glance.

The move: a three-corner carousel

Turn the whole cube so the corner that's already home sits at the top right front. Then:

U R U' L' U R' U' L

That corner stays put, and the other three ride around in a loop, counterclockwise as seen from above. Here one pass is enough: all four corners are in their places. True, they're still twisted this way and that, but that's a job for the next chapter.

Notice that the algorithm has four turns of the top, two one way and two back, so overall the top goes nowhere. The yellow cross edges you lined up last chapter end up exactly where they were.

Sometimes twice

Three corners can circle in either direction, but our algorithm only sends them one way. If the corners need to go the other way, one pass won't do it: afterward they're still out of place. No problem. Play the algorithm again: moving two places counterclockwise is the same as moving one place clockwise.

That's exactly the case here: after the first pass the corners are still in the wrong spots; after the second, they're home.

Why only three corners move

The algorithm follows a pattern: do A, do B, undo A, undo B. Here A is U R U' and B is L'. A sequence built this way is called a commutator.

The trick is that A and B barely get in each other's way. U R U' works on the right and at the back, L' on the left. They share just one piece: the top left back corner. Anything only one of them touches is put back when that move is undone. And because of that single shared piece, exactly three corners end up going around in a loop.

Commutators are the chief tool of people who invent algorithms. The sexy move is one too, by the way: A = R, B = U. But those two moves share more pieces, which is why the sexy move disturbs seven pieces rather than three.

Once you know the principle, you can compose algorithms of your own: find two actions that overlap in just one or two pieces, and string them together as "there, there, back, back."

No corner in place

Sometimes not a single corner is home. In that case, play the algorithm once with the cube held any way you like. Afterward, exactly one corner is guaranteed to be in its place.

Here it's the top right back corner. Rotate the whole cube with y to bring it to the front right, and carry on as usual.

What you'll never see is exactly two or three corners in place. That's the third law from chapter 1 at work: you can't swap just two pieces. The top edges are already right, so the corners can only be mixed up "fairly": all four in two swapped pairs, or three in a loop. That's why you'll always find no corners in place, one, or all four.

Where people trip up

• Checking a corner by one sticker. Yellow on top doesn't mean a thing yet. Match all three colors against the three centers.
• Turning the top on its own. A stray U knocks the solved edges out of line. To bring a corner to the front, turn the whole cube.
• Holding the wrong corner. The corner at the front right must be one that's already home. Otherwise the algorithm will lovingly guard a wrong corner while sending the others around in pointless circles.
• Expecting the corners to come out straight. They won't: a twisted corner sitting in its own spot counts as a success.

Try it yourself

Put all four corners in their places. First find the corner that's already home; there's only one here, and it already sits at the front right. Then play the algorithm as many times as it takes.

Next: the final twist

All the corners are in their places, but some still have their yellow facing sideways. One step remains: twist them where they stand.

The move for this looks like madness. Halfway through, the cube falls almost completely apart. But then it puts itself back together, and in the next chapter we'll see why it simply has no other choice.