5x5 to 7x7: centres, edges and parity

Bigger cubes are solved by reduction, as the 4x4: build the centres, pair each edge's pieces, then solve the cube as a 3x3 with outer turns. Odd cubes (5x5, 7x7) have a fixed centre piece on each face and a fixed middle edge piece in each edge, so the colour scheme is given and each edge is paired around its middle piece; even cubes (6x6) behave like the 4x4 and have its parities.

Centres are built from bars as wide as the centre, joined row by row: first two opposite centres, then the side centres around them. On the 6x6 and 7x7 the bars are longer and the order matters more, so solvers plan a whole bar at a time. The last two centres share their pieces, and their cases are the L2C sets.

Freeslice pairs the edges with the slices left out of line: rather than restoring the centres after each pairing, keep working with the slice turned and restore it only when the last edges are done. It is common among the fastest 5x5 and bigger-cube solvers. Pairing ends with two edges left, the L2E sets.

On odd cubes the last two edges can show edge parity: a middle piece flipped against its wings, a position no 3x3 has, fixed by the parity cases of the L2E set. Even cubes add the 4x4's parities, a single flipped dedge or two swapped, for the 3x3 stage.

Yau's order works on every size: two opposite centres, three cross edges built and placed, the last four centres, then the remaining edges and the 3x3 stage. It has spread from the 4x4 to the 6x6 and 7x7, where building the cross edges while the cube still has room saves time in the crowded last stages. Hoya's order (the cross before the last two centres) is also used on the 5x5.

The 6x6, like the 4x4, has no fixed centres or middle edges, so a reduced 6x6 can show OLL and PLL parity. It is fixed with parity algorithms like the 4x4's, turning the inner slices.

Cubes beyond the 7x7 are solved the same way, with longer bars and more edge pieces per edge: even sizes behave like the 6x6 and odd sizes like the 7x7. No algorithm sets are published for them; the L2C and L2E ideas carry over, applied to the inner slices that differ.

Once reduced, the cube is solved with outer turns as a 3x3, with the 3x3's methods and algorithms. On the 7x7 the 3x3 stage is a small share of the solve; on the 5x5 it is larger, and a good last-layer system pays off.