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Is there a website where I can find an explanation for the rules of FMC and how it works?
Because I see all these things like premoves and insertions and other things I've never heard about, and I'd really like to know what it all means

Caseyd: a cancellation is when moves of parts of your solution cancel out.

example OLL - PLL:
OLL Sune: R U R' U R U2 R'
PLL (any): anything that starts with an R, R' or R2 will cancel one or more moves with the last move of the Sune
Examples:
CClockwise U perm: R2 U R U R’ U’ etc. cancels 1 move: R’ followed by R2 = R
Clockwise U-perm: R U' R U R U etc. cancels no less than three moves!
R U R' U R U2 R' R U' R U R U = R U R' U R U R U R U
R' and R cancel out and then U2 and U' is the same as U
= 3 cancelled moves

Spotting cancellations is one of the reasons for writing your FMC solutions without cube rotations.

Suppose you finish the 3rd pair with R’ U2 R and do the 4th with y’ F’ R U R’ U’ R’ F R to keep EO:
y’ F’ = R’ so you are actually cancelling two moves.
However if you write your results as R’ U2 R y’ F’ R U R’ U’ R’ F R you’ll score 11 HTM for this part
Still 11 HTM if you would write R’ U2 R R’ B U B’ U’ B’ R B in your final FMC solution ;-)
But only 9 HTM if you cancel out the R R’ and write R’ U2 B U B’ U’ B’ R B

Other cancellation tricks:
Watch out when you consecutive moves of opposite faces:
… R U R’ followed by L R UR… = … R U R’ L R UR… = … R U R’ R L UR… = ... R U L UR…
And pay attention to AUF’s too
AUF before a PLL could cost you a cancellation with the end of OLL
AUF before a PLL that ends with U is just a silly way of increasing movecount

I used OLL and PLL as examples (even though you typically would avoid ending with PLL in FMC) but offcourse they can be replaced by any moves in any part of the solution

Some of the parts seem a little manual, but the cube comes together well in the middle. More importantly: Why in the world would you AUF twice during the LL?

For starters, learn to take advantage of symmetry. Why did you use that OLL when its mirror leaves a J-Perm instead of a G-Perm? That'll save 2 moves anyway, assuming you know the optimal J-Perm.

EDIT:
Incidently, I note that the first G-Perm I learned actually gets 3 moves to cancel on that solve. Knowing extra algs for the same case can be beneficial. But generally OLL/PLL is not the most efficient way to finish solves for FMC.

Spoiler

Scramble: R2 D' L' U2 B2 D2 B' U L2 D' U F' R' B2 U2 F U D' L2 R U' F2 L' B' F'
JackJ's start:
D' R2 D2 R' D2 R2
U2 L U B2 U L2 U' L2
R' U' R U2 R B R' B U B' U' B
L2 D L' U2 L D' L' U2 L'
Alternate G-Perm finish (3 moves cancel):
L U' R U2 L' U R' B' F' U2 B F U (48-3 = 45)

D' R2 D2 R' D2 R2
U F' L2 F U B2
U B U' B' R' U2 R
R U' L U2 R' U R U2 L' R'
B2 L2 (30)

If you don't know about pseudoblocks and premoves you should; they can be very useful. Try to find the relation between the pre moves (B2 L2) and how the cube looks after 12 moves. The premoves are then applied before the scramble to make the F2L-1 look normal and when you're done with the solve you just slap them on right at the end.

The best way to solve the LL is for the most part Snyder's approach (leave three corners with a short alg) though you don't need to instantly which alg to use just muck about a bit. Leaving three corners Heise style is also really useful.

btw I saw a post by (I think) Mirek on the Yahoo group saying that all 10-move 2c2e swaps (e.g. J-perm) are cyclical shifts of each other, but I can't seem to find it now. I feel a bit stupid for not noticing that the three algs I know (that affect only one layer) weren't shifts of each other, oh well.

Has anyone made any FMC videos? Like a time lapse or a "how to" demonstrating different methods and tricks. I am interested in getting into it and somethings are hard to understand through reading.

I have received numerous requests for a fewest moves video tutorial. I will never make one. I don't see what there is to be gained through video that would actually be useful to anyone.

Here's some movecounts that I gathered from experience, imo it's very useful to know what's a good start and how many moves you can expect to finish up in. Feel free to correct and such.

Am I right in thinking that for the 2-corner twist you insert a random 3 corner cycle, then insert another one? 6 cancelled moves sounds a lot for one insertion, although I guess there are a lot of possibilities.

Wow, irontwig, I wish I were that good. Almost everything there is beyond me. I have to admit that if I ever get one move more than what you have for any of those things, I consider myself lucky.

But I also admit that the ratio of those values seems similar to what I see, except that I don't see how you manage to get such good results with 2-corner twists. I generally have trouble doing better than 14, so I just apply the algorithm for it at the end when I get them. How do you manage it?

I assume the higher-than-optimal average for edge 3-cycles is due to not always finding the edges on the same slice and having to resort to 9 move U-perms?
Seems you did a -2 vs the optimal moves, perhaps the edges could get an average of 6 because sometimes you will find them on the same slice with cancellations to boot.

What HTM wold you consider the max for "leaving 5 corners"?

For three edges most of time the best is either a 7-mover cancelling no moves or a 10-mover (e.g. [R E R',U] or [R U R', S]) cancelling three moves. There are only two non-isomorphic 6 and 8-mover so of course those are rarer.

For the two corner twists there often several three move conjugates such L F2 L' or D R D' in the skeleton and you only need to be able to cancel away one of those to be able to leave three corners with only two extra moves.

For the two corner twists there often several three move conjugates such L F2 L' or D R D' in the skeleton and you only need to be able to cancel away one of those to be able to leave three corners with only two extra moves.