This is the method i use for the 2x2x2.
The 1st step:
First we orient all corners on D side. This can be done in any order The permutation will be done in the 3rd step. I have placed a few examples for the first side but you can invent them on your own for diffrent cases. Most of the time the scramble gives you 2 or 3 corners already oriented. If you have non try to see the fastest 2 corners and go from there to see what kind of case you will be in. Try to see in what order the corners are on the D side. This is usefull when your done with the 2nd step.
The 2nd step:
The orientation of the D side. For this also the same thing as the 1st step. The corners can be in any order. The permutation will be done in the 3rd step. When you are done with the 2nd step and know in what order the D side is you only have to look at the U layer and can easy start with the 3rd step.
The 3rd step:
We permute all the corner of U and D side.
1st Step Orient D Side
The Red square or line are on U layer/side and 3 corners oriented on D side.
U R2 U' R2
R' U R
R U' R'
The red line is on D layer and 3 corners are oriented on D side.
F (R U' R')
(R' U)(R U R')
The red lines are on D layer and 2 corners oriented on D side.
R' U2 (R U' R')
(R' U)(R' U R')
(R U') R2 U R'
R U2 R2 U R2
2nd Step Orient U Side
(R' F) (R U) F U' F'
R' U' R U' R' U2 R
(R U R' U') (R' F R F')
R U R' U R U2 R'
F (R U R' U') F'
(R U') (R2' F) (R2 U' R')
Alg change the order of the D side.
R2 U2 R U2 R2
3rd Step Permute U and D
R2 F2 R2
R2 B2 R2
R2 U R2 U' R2 U R2 U' R2
y R' D L' U2 L D' R
R2 D R2 D' R2 D R2 D' R2
y R U' L D2 L' U R'
R2 U R2 U2 F2 U F2
y2 R2 U' R2 U2 B2 U' B2
