I want to share with everybody a 27 HTM corner-first FMC solve I got a couple of days ago. I've found it pretty quickly during the 1 hour time limit and than I've "submitted" a 25 HTM "standard" solution. Anyway, it's nice and quite simple to understand even if you are not familiar with corner-first solves.
The scramble is taken from the German forum weekly competition, even though from which week since I haven't fetched it myself.
Scramble: U' F2 U F2 D R2 B2 D2 F2 D2 U' B' L U' B2 F2 R2 D L' R
Solution: L' F' D R D2 B D' U R' D2 F' D' R2 D R' D' R D U' B' D B' U' L2 D2 B2 U2
L' F' on normal scramble orient corners. Solving corners by orienting first and then permuting not only gets you (usually) short solutions, but makes it somehow easy to solve edges in-between the two steps. I haven't exploited this advantage now, however. Note how with these 2 moves I also make one 2x2x1 square and 3 CE pairs (we can say that there are 5 edges solved in this case).
Then I switch to inverse scramble with premoves F L. Then U2 B2 D2 L2 U B2 D' B2 solves the permutation of corners and one more edge.
There are now 6 edges that need to be cycled. The permutation can be broken into 2 disjoint cycles:
UB->BR(->BU)
LF->FR->LD->FD(->FL)
The two cycles are both odd (made up by an even number of pieces; yeah I know, the terminology is confusing) and "flipped", which means I can't solve them separately; actually, one of the two conditions (either odd or flipped) is enough. However, as you will see, if we allow centres to be unsolved, we can solve an odd permutation (especially 4-cycles) on its own.
Disregarding centres (if at the end they are not solved, you can insert M E M' E' or similar moves somewhere and cancel lots of them, usually), we can cycle 4 edges by setup + single slice move + anti-setup. Before the last 3 moves (B2 D' B2) it's easy enough to solve the cycle UB->BR->LF->FR. This solves two more edges (FR and BR) and leaves the other 4 in a 4-cycle, which we can then solve with a similar technique.
How to solve them?
Setup: B D' B
Permute: E (or U D' y' if you don't know which way is E and which E')
Anti-setup: B' D B'
And one move even cancel with the B2 D' B2. Pay attention when re-writing the solution without rotations or slice moves (you can use something like Uw' U if you prefer). The solution up to now looks like:
On inverse scramble with premoves F L
U2 B2 D2 L2 U //Partial CP (3 moves left undone)
B D' B U D' R' D R D' R2 //Leaves 4 corners and 4 centres
Then we can solve the last 8 pieces:
Setup: D F D2 R
Solve: E'
Anti-setup: R' D2 F' D'
Which turns out to look like:
D F D2 R U' D B' D2 R' D'
And the cube is solved. Thanks to Attila for sharing many of his corner-first solution
pb by 5 moves. Will post scramble and solution here once I get home 
So i am ok with the 26.
