Shortcuts

2x2 ("Pocket Cube")

Case #1

R' U' R U' R' U2 R

Case #2

L U L' U L U2 L'

Case #3

R2 U2 R U2 R2

Case #4

F [R U R' U'] [R U R' U'] F'

Case #5

F [R U R' U'] F'

Case #6

[R U R' U'] [R' F R F']

Case #7

[F R U' R' U' R U R' F']

After performing the above step, if one side is solved orient that side to the right (as seen below) then perform a U' before using the following algorithm. If opposite corners are matched do the algorithm to get a matching side, then perform a U' followed by the algorithm again.

(If side solved) U', then:

L' U R' D2 R U' R' D2 R2

2x2 info originally from rubiksplace.com.

Megaminx

Solve all but the last layer "intuitively" using simple movements to move sides and corners and then slot them into place. Great video teaching this approach to the Megaminx: YouTube.

Last layer algorithms (if using the alt version of the corner orientation, this is just like solving the last layer on the 3x3):

Megaminx info from the above linked YouTube video.