First off, grab a 2x2 cube. (Do NOT use a 3x3 right now, it will be harder)
Then, apply the sequence R2 U F2 U' F2 R2. (Or you could do any other sequence that also preserves CO)
Now, do R U R'. You will notice that you have 3 corners that are not oriented. You can use moves that preserve CO to set your 3 'bad' corners up to this case.
For example, solve your 2x2 cube and apply R' U' R U' R' U2 R. This is an antisune, but you can definitely orient the corners in less than 7. Notice that you can do R2 U' to set up the 3 corner / R U' R' case, and then simply do the R U' R' trigger. So, your CO solution would be R2 U' R U' R' (5).
Are you still following me?
Haha I hope so.
If you have 3 misoriented corners, you can also use the trigger R U R'. So R U R' / R U' R' both work to setup 3 bad, but also they are literally just different by an R2 move so not very different. Again, use the moveset that preserves CO (R2, U, F2) to set these up.
Now, there is also the R U2 R' trigger for 4 corners and the single move trigger R, also for 4 corners.
Yes, you can orient 4 corners in 1 move. I won't get into as much detail for these because you know the drill by now. Set up and trigger.
Try these setup exercises for 4 bad corners:
Apply R2 U' F2 U2 R U2 R' U2 L2 U2. Solve CO on white/yellow in only 4 moves by setting up to and using the R U2 R' trigger.
Apply F2 U R2 U' F2 R' U2 F2 U' R2. Solve CO on white/yellow in only 3 moves by setting up to using the single move R trigger.
Now, here are some random 2x2 scrambles and efficient CO examples.
R' F U' F U R F' U2 F ----- y' U2 R2 U R // CO (4)
U' F2 U2 F2 R F' R2 F' R ----- z x R2 U2 R // CO (3)
F' R' U2 F' U F R U2 F ----- y z R U' R' // CO (3)
U F' R2 U' R U R' F' R2 ----- y F R U' R' // CO (4)
Very useful tip: If it isn't easy to set things up, just look at a different axis / colors and most likely it will be a much easier case.
If you have any questions then feel free to ask!
