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3x3: Fridrich full oll+pll
3Fewest Moves: Mixture depending on context. Either optimised fridrich, heise or petrus combined with intuitive blockbuilding.
4x4: Reduction
4Fewest Moves: 2x2 reduction/intuitive blockbuilding
Megaminx: cross+f2l+blockbuild+f2l+commutator

2x2: First layer+OLL/PLL Haven't bothered to learn any other method.
3x3: Fridrich F2l with 3Look LL.
4x4: IDK what its called. Centers, edges, solve like 3x3.
5x5: Same as 4x4.

2x2x2: First layer, LL: sometimes fridrich, sometimes 2 alg-method (dunno the axact name)
3x3x3: Intuitive Fridrich F2L, then 2 Look OLL, then 2 Look PLL (CFOP)
4x4x4: Intuitive centers and edges, then CFOP (same like 3x3x3)

2x2 - Whichever method that'll get the job done fastest, but usually ortega.
3x3 - Fridrich (3LLL) - About to start learning all my OLLs =]
4x4 + 5x5 - Reduction
6x6 + 7x7 - Reduction as well

2 - SS
3 - MGLS-F
4+ - Reduction (centers [2-4-6-9 on 5x5, cols for 6+], AvG, MGLS-F)
BLD - 3OP (L2R2)
Megaminx - normal
Pyraminx - Wanna do Petrus, will probably end up using Cooper
Square-1 - Something kinda SSS1M
Clock - Pochmann
Magics - Pochmann

2x2: LBL, ortega, luck
3x3: Cross-Slots, sometimes random blocks for F2L. VHF2L-COLL-EPLL or OLL-PLL.
4x4: opp centers, other centers, 6 edges on M slice, 2 edges, 3x3, parity fix.
5x5 and larger : Same as 4x4, line approach for centers and one tredge at a time.
Pyraminx: my own methods. FL: tips-edges or Japanese. LL: 1LLL. Learning FP.
Megaminx: intuitive F2Ls, Y, N, and T perm edges, corner commutators.
SQ1: intuitive everything until LL. J perm, Uccw, parity fix.

2x2: Ortega
3x3: Fridrich
4x4: Reduction(2-2-2-2 edges on M slice)
5x5: Reduction(Freeslice edges on E slice)
6+: Reduction(1x4/5 centers, Freeslice edges on E slice)
Clock: My own method
Sq-1: Group corners for cubeshape, minimum algs for the rest.
Pyraminx: LBL

2x2: Ortega, CLL, LBL, a bit of EG
3x3: cross on left Fridrich
4x4: reduction, FR edges, opp color neutral fridrich
5x5: reduction, FR edges, opp color neutral fridrich
6x6: reduction, Multi-redux for edges using FR, opp color neutral fridrich
7x7: reduction, Freeslice then FR for last 4, opp color neutral fridrich
BLD: M2, pochmann corners.
Sq-1: Vandenbergh with few algs.
Pyra: Polish V
Clock: semi-freestyle Pochmann.
FMC: petrus start, sometimes EO, finish skeleton to either 5 or 3 corners, insert commutators to cancel. or anything else that looks good.

2x2: first side (NOT LAYER!) -> Oll -> finish the rest.... otherwise known as ortega
3x3: Intuitive f2l -> Oll (sometimes 2-look, mostly 1 look because I still need to learn like 10 of them...) -> PLL 1-look... also known as the Fridrich Method
4x4: Centers-> pair edges (2 pair) -> finish like the 3x3 with parity.... known as reduction method...
5x5: same as 4x4
6x6+ (if I had v-cubes =b): Assuming I had them... I would prob. use redux... like the 4/5x5.
Pyraminx (broken, unfortunately): the awesome method eric limeback teaches you on youtube...
megaminx (I don't have one but my friend does and he lent it to me for a week...): the method Erik teaches you.
Square-one: The one taught by elimescube or somethin.... pretty much it's make it cube'd -> solve corners on one side (one will be permuted and the other will only be orientated) -> orient the edges -> fix the non-permuted corners-> permute corners.

2x2: primarily Guimond, 'tho I sometimes use Ortega if I can see the PLL or LBL in the hopes of doing CLL.
3x3: Fridrich
4x4: reduction, Syuhei style (cube rotations instead of D inserts)
5x5: reduction, centers generally by 2,4,6,9, freeslice edges on E
6x6: solved it once (don't own one), used reduction, centers by middle 2x2, extend to central 2x4, 3x4, 4x4, L2C by central 1x4, extend to left 2x4, 3x4, 4x4, freeslice edges on E
7x7: reduction, centers by middle 3x5 and extend, L2C by the one all the good cubers use, freeslice edges on E
3BLD: 3OP
4BLD: most likely commutator centers, r2 edges with commutator algs for any l/r cases, Old Pochmann corners
Megaminx: all but LL using the Erik guide, LL using cutex.info algs
Square-1: Vandenbergh with lots of algs missing
Pyraminx: CF

2x2: I don't solve this one a lot, but I'd probably do layer by layer....
3x3: ZZ Winter Variation (EOLine, F2L + corner orientation simultaneously, PLL)
4x4 and up: Direct solving method (Solve columns of pieces until you have a last layer, then use lots of commutators. I don't speed solve these puzzles!)