Putting the corners in place

The last four pieces

Everything is solved except the four top corners. They're wrong in two ways at once: they sit in the wrong spots, and their yellow points wherever it likes. 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 place. Work with the algorithm alone, 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 if its three colors are the same as the three centers around it. Which way the 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. Around it sit the yellow, green and orange centers. They match, so the corner is home — even though its yellow sticker is facing forward. The other three corners here, on the other hand, are not in their places. Check each one.

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

The move: three corners round a circle

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 travel round a circle — counterclockwise, seen from above. Here once is enough: all four corners are in their places. True, they're still twisted every which way, 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 all in all the top goes nowhere. The yellow cross edges we lined up last chapter end up exactly where they were.

Sometimes twice

Three corners can travel round a circle in two directions, but our algorithm only sends them one way. If the corners need to go the other way, one pass won't do it: afterwards they're still out of place. No problem — play the algorithm again. Two laps counterclockwise are the same as one lap clockwise.

That's exactly the case here: after the first pass the corners are 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 construction like this is called a commutator.

The trick is that A and B hardly 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 that only one of the two touches gets undone when that move is played backwards. And because of that single shared piece, exactly three corners end up going round a circle.

Commutators are the main tool of people who invent algorithms. The sexy move is a commutator too, by the way: A = R, B = U. Only there the two moves share more pieces, which is why it disturbs seven pieces rather than three.

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

No corner in place

Sometimes not a single corner is home. Then play the algorithm once, holding the cube any way you like. Afterwards, 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: 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 round a circle. That's why there are always either no corners in place, or 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 separately from the algorithm. A stray U knocks the solved edges out of place. To bring a corner to the front, turn the whole cube.
• Holding the wrong corner. The corner at front right has to be one that's already home. Otherwise the algorithm will lovingly guard the one wrong corner and send the others round in circles for nothing.
• Expecting the corners to come out straight. They won't: a twisted corner sitting in its own spot is 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's already at front right. Then the algorithm, as many times as it takes.

Next: the final twist

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

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