4x4: centres, edges and parity

A 4x4 has no fixed centres and no fixed edges. Every speed method reduces it to a 3x3: it solves the centres (four pieces each), pairs the 24 edge pieces into 12 'dedges', then solves the result with outer turns. The methods differ in the order: Yau, the most used, builds part of the cross before the last centres; plain reduction finishes every centre first; Hoya places the whole cross before the last centres.

R turns the right face alone; Rw (or r) turns the outer two layers together; 2R turns the inner layer next to R alone; x, y and z turn the whole cube. Inner-layer turns move centres and wings, so they build the reduction, and the outer turns that finish it are the 3x3's.

Centres are built from bars, two centre pieces side by side, joined into a full centre with an inner-layer turn and moved out of the way with an outer turn. Start with two opposite centres, then build the side centres around them. Because the centres are not fixed, they must be placed in the colour scheme: opposite colours opposite, and the side colours in the order the corners need. The last two centres share their pieces; a few algorithms handle the patterns that are slow to see.

Named after Robert Yau, who made it popular, Yau is the standard 4x4 method among fast solvers: two opposite centres (usually the cross colour and its opposite), three cross dedges paired and placed around the bottom centre, the last four centres, the last cross dedge, then the remaining edges, and the 3x3 stage. Solving three cross dedges early, while there is still room to move edges freely, means the cross is already done when the 3x3 stage starts, and the last centres are built in the space the cross leaves.

With the centres done, the remaining dedges are paired a few at a time: line up wings that belong together on opposite sides of a slice, turn the slice to join them, replace the joined dedges with unpaired ones using an outer-turn sequence that keeps the joined pairs safe, and turn the slice back so the centres are restored. The usual grouping after Yau is 3-2-3: three dedges at once, then two, then three. Plain reduction often pairs six, then the last two (6-2).

When two dedges are left with their wings split between them, one algorithm pairs both without breaking the centres or the other dedges.

Because the centres are not fixed, a reduced 4x4 can show the two cases a 3x3 never has: one dedge flipped (OLL parity), or two dedges swapped (PLL parity). They come from the inner slices and need inner-slice algorithms. Recognise OLL parity by an odd number of flipped edges during OLL, and PLL parity by a last layer that matches no PLL.

Instead of fixing OLL parity and then doing an OLL, the one-look sets do both at once: every corner orientation with one flipped dedge, and with three. They save a look and many moves, at the cost of a large set to learn.

Jong-Ho Jeong's Hoya builds the centres but two with the cross in mind, places the cross dedges, then finishes the last two centres and pairs the last eight edges before the 3x3 stage. It trades Yau's free space for the last centres for a cross that is complete earlier.

Learn the parity algorithms first, since most solves need at least one, then drill centres and edge pairing on their own: the Solve desk's drill picker has the 4x4's centres, edge pairing and Yau starts. Look ahead during the centres for the next bar, and during edge pairing for the next matching wings, rather than for speed of turning.