Learn · the 4×4
One more layer, two new problems
A 4×4 has no fixed centres and every edge is split in two, so the method is to rebuild it into something that behaves like a 3×3 and then solve that. Almost everything you already know still works.
The first thing to know about a 4×4 is that it is mostly a 3×3 in disguise. The last two thirds of the solve are the method you already have. What changes is the beginning, and it changes because of two missing certainties.
On a 3×3 the centre of each face never moves. Red is opposite orange because red and orange are joined through the middle of the puzzle, and no amount of turning alters that. A 4×4 has no single centre piece at all: each face carries 4 centre pieces that travel like anything else. Nothing tells you which colour belongs where until you decide, so the colours opposite each other are yours to establish.
The second missing certainty is that a 4×4 edge is not one piece. Each of the twelve edges is a pair, so there are 24 edge pieces in all, which cubers call wings, sitting in 12 pairs. The corners are the only pieces that behave exactly as they always did: there are still 8 of them, still with three stickers each.
8 pieces
Corners
Three stickers each, and each one unique. These behave exactly as they do on a 3×3.
24 pieces
Wings
Two stickers each, in 12 matching pairs. Join each pair and the puzzle grows twelve edges.
24 pieces
Centres
4 to a face, all four the same colour, all of them free to travel.
Reading the notation
An extra layer needs a way of naming it. R still means the right-hand face on its own. Rw, which some write as r, means the right face together with the layer behind it, two layers moving as one. And 2R means that second layer travelling alone, leaving the outer face where it is. That last one is the move a 3×3 does not have, and everything strange about a 4×4 comes from it.
Watch the difference. The player on the right turns the outer face and then the inner layer on its own, twice each, and the two are plainly not the same move.
The method: reduce, then solve
One. Build the centres. Make each face's 4 centre pieces match, one face at a time, using inner-layer turns to slide pieces into place. There is nothing to memorise here, and looking for an algorithm will only slow you down: the pieces are interchangeable, so almost any sensible move works. Do keep the standard colour arrangement, white opposite yellow, green opposite blue, red opposite orange, because the rest of the solve assumes it.
The one moment that stops beginners is the last two centres, when finishing one seems to break the other. This is the sequence for it: 2R2 2U2 2R2 2U2. Two inner-layer half turns, alternated. It moves 4 centre pieces and touches nothing else at all , no corner and no wing, so you can use it whenever you like without consequences. Run it three times and you are exactly back where you started, which makes it safe to experiment with.
Two. Pair the wings. Bring each pair of matching wings together so the puzzle has twelve whole edges. Most solvers do this by lining two wings up in the same layer and then turning the outer face between them, and there is a standard trigger for it, Uw' R U R' Uw. Looking will get you further here than memorising. Note that the trigger disturbs the layers below, which is exactly why the pairing is finished before the 3×3 stage begins and not during it.
Three. Solve it as a 3×3. Treat each joined pair as a single edge and each centre block as a fixed centre, and the puzzle is a 3×3. Every method you know applies without alteration: the cross, the first two layers, the last layer. If you have worked through the beginner's method or the full system, use it here unchanged.
And then, about half the time, the 3×3 stage will present something a 3×3 cannot do: one edge turned the wrong way round, or two edges that need swapping and nothing else out of place. No last-layer algorithm will touch either. This is parity, it is not a mistake you have made, and it has a page of its own.
The 4x4 engine behind these pages is checked against the 3x3 engine on every move the two puzzles share, so a disagreement between them fails the build. Both parity algorithms are then measured rather than trusted: each is applied to a solved cube and every piece is compared with where it belongs, and each is confirmed to undo itself. The animations run on a separate engine written in JavaScript, held to the same answers by the same tests, so the cube a reader watches cannot contradict the cube that did the checking.