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First fix the center planes; the first 2 opposite, then adjacent. Then fix the edges (excl. corners). After that the big cubes behave like the standard 3x3.

Front/right Edge flip R U R' F R' F' R *** (Edge parity Uw' ... Uw)

In the 4x4 cube some strange configurations/notations can happen:

Make sure that the face centers are correctly aligned. Not mirrored. To exchange a row or a facelet between planes, level up the target and the donor, then do a halfturn in the donor plane. Move the target row into the donor plane, do a halfturn again, and restore.

To fix the Edges, hold the edge cells to the front left/right sides. They should not face with the same color when on opposite sides. Bring them together and save to the top face. Bring a broken edge back down, restore centers.
Last edges: make sure the matching edge pieces are level (front). Bring one over to the other edge. Flip*** and restore.

OLL parity, see general video below Rw U2 X Rw U2 Rw U2 Rw' U2 Lw U2 Rw' U2 Rw U2 Rw' U2 Rw'

PLL parity, see general video below 2R2 U2 2R2 Uw2 2R2 2U2 ** -- switched edges across front/back. note: 2R2 is replaced with Rw2 R2 for now.
Other case: R U R' U' (as above) U R U' R' -- switched edges neigboring front/right.

Notation 2R or lowercase r means only turn the second layer from the right (no button yet); Sometimes lowercase r or Rr are alternative notations for Rw but some use r = R2

General solving of 4x4 cube: ©J Perm

OLL parity intuitively: ©SpeedCuberGuy -- (2R U2)*4 2R, then redo edges, etc... -- note: 2R is replaced with Rw R' for now.

PLL parity intuitively: ©SpeedCuberGuy -- see algorithm** above

PLL parity is also 2 wrong positioned corners remaining, else fixed. Hold one such corner to the front/up/left, the other to the up/right, and apply algorithm**. Then fix the edges, and corners of the last layer again. ©Cube Solve Hero

Centers: Work with rows of 3, first the middle one. Start with the middle cells.
To exchange an excentric row or a facelet between adjacent planes, level up the target and the donor, then do a halfturn in the donor plane. Move the target row into the donor plane, do a halfturn again, and restore.

Last 2 Centers (L2C): Hold the 2 unfixed planes in front of you (F and U planes), Mirror the below cells in the F/U edge:
To switch the right cross cells: Rw' F' Lw' Rw U Rw U' Lw Rw'
To switch the lower right point cell on the up plane with the upper right point cell of the front plane: Rw U Rw' U Rw U2 Rw'

Last 2 edges can be done one edge at a time by edge flip*** and "single twisted edge" algorithms: ©Cubeverse Tutorials
Hold the last 2 broken edges to the front left/right initially (manipulate to be flipped*** edge to the front/right if necessary). Moves are equator parallel.
In both below cases the "last" single piece should be **diagonally on the opposite side from where it must go.
A block of 2 untwisted edge pieces with the "last" piece **across: Bring the other single piece, at the position where the "last" piece must go, over to the edge with the "last" piece, and flip*** that edge. Restore centers.
Or with a twisted block, when you move over the single** in/to the double, an untwisted block should be formed. flip*** the edge with this block. Restore centers.
When you have one twisted edge remaining do the "single twisted edge" algorithm.
When you have all the pieces at the same side for both broken edges, do the above steps to get one of the above starting situations. Or do the single twisted edge algorithm for each edge.
other flipping algorithm mentioned in the video is R' F D'

For the last 2 edges: ©CubeSkills   ©Sarah.cubing.net   ©SpeedcubeDb   General solving of 5x5 cube: ©Noah Richardson or ©J Perm

General solving of 6x6 cube: ©J Perm

Centers: work in rows, start with center row, center cell.

Last 2 Centers (L2C): first do the central row, then the adjacent one, then the distant one at that side, then the adjacent from the other side, and the other distant one.

Last 2 Centers (L2C):

Switch lower right point cell (=center plane corner) Up plane with upper right point cell Front plane: Rw U Rw' U Rw U2 Rw' -- note: mirrored position.

Switch second cell from the back in the right column of the Up plane center with the same cell position of the Front plane: Rw' F' 3L F Rw F' 3L' (F)
Same for the left column: Lw F 3R' F'Lw'F 3R (F')
Same but second from the left in the top row: 3L F Rw' F' 3L' F Rw (F')

Middle piece and rest of row are both in workplane (up face), but opposite: M U2 M'
 Piece(s) and rest of row are both in workplane (up face), but opposite. Hide piece slice(s) in back plane, U2, and restore. Slice(s) must be in same distance from center, not to break the back plane.

Last 2 Edges: hold them to the top/front and top/back. Middle piece is leading to where the colors must go. To exchange pieces mirror them in the line from left to right. To flip the front edge F U' R U

All exchanges are combinations of the following with clever flips and U-turns:

Parity and Orientation:

You can solve this per the last 2 edges or per one. Middle piece is leading. For one edge to the top/front:

under construction, incomplete instructions.

6x6 PLL parity fix: 3Uw2 (3Rw2 R2) U2 (3Rw2 R2) U2 (3Rw2 R2) 3Uw2 --switched edges across front/back
Other case: R U R' U' (as above) U R U' R' --switched edges neighboring front/right, additional sexy move and undo

Edge parity fix, last edge twisted: 2R' U2 2L F2 2L' F2 2R2 U2 2R U2 2R' U2 F2 2R2 F2

General algorithms

Corner twisting: R' D' R D. Keep the lower layers of the cube steady.

Corner cycling: top view   ↓←↗.   U R U' L' U R' U' L

OLL edges, fix cross:   . 🠢 ⅃ 🠢 __ 🠢 +   F R U R' U' F'

Exchange 2 edges:   L or __ correct   ⤡L   R U R' U R U2 R' U   Note: not necessary for PLL

OLL corners. algorithm   R U R' U R U2 R'

PLL corner side colors to the same side, T Perm   R U R' U' R' F R2 U' R' U' R U R' F'

PLL edges to match the corners. U Perm   R U' R U R U R U' R' U' R2

Front/right Edge flip R U R' F R' F' R

Switch lower right point cell (=center plane corner) Up plane with upper right point cell Front plane: Rw U Rw' U Rw U2 Rw'