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Because of the prominence of DR as an FMC method, the most important algs would be the DR 3e and 2e2e algorithms.
(R2 U R2 f2)2
(R2 u R2 f2)2
(R2 U r2 F2)2
(R2 u r2 F2)2
M' U2 M U2
(R2 U2 R2 F2)2
F2 R2 U2 F2 U2 R2 F2 U2
L R U2 L' R' F2 L2 F2
(U R2 U' M2)2
(M2 U2)2
(R2 U2)3
R2 U2 L2 U2 R2 D2
(M E2)2
L R U2 L' R' F' B' U2 F B
R2 S2 R2 U R2 S2 R2 U'
and probably a few more. Keep in mind that slice moves are not allowed in FMC, I just wrote them that way for easier memorization.

Because of the prominence of DR as an FMC method, the most important algs would be the DR 3e and 2e2e algorithms.
(R2 U R2 f2)2
(R2 u R2 f2)2
(R2 U r2 F2)2
(R2 u r2 F2)2
M' U2 M U2
(R2 U2 R2 F2)2
F2 R2 U2 F2 U2 R2 F2 U2
L R U2 L' R' F2 L2 F2
(U R2 U' M2)2
(M2 U2)2
(R2 U2)3
R2 U2 L2 U2 R2 D2
(M E2)2
L R U2 L' R' F' B' U2 F B
R2 S2 R2 U R2 S2 R2 U'
and probably a few more. Keep in mind that slice moves are not allowed in FMC, I just wrote them that way for easier memorization.

Would the algs from this thread also be useful? I'd assume that HTA is a natural progression from DR or is it generally considered to be too linear ? =>

Second step: equator
here is how I do this step: I find/place a pair of edges that need to be fixed onto the L/R slice (I use F and B for the EO so I am free to turn L and R in the next step). Next I find a U/D color and place it above the pair, and place the opposite color below. now when you place the two U/D edges it also partially solves your equator. usually doing this you either end up with a nearly solved E slice or solved.

Fourth step: corner permutation
There are only 5 cases, I generated my own algs for these but it would be very easy to generate others if you do not care for these:
if D is solved do Y or J perms on U, else you will have one of these 3:

Double J: (place the correct corners on L)
F2 U' R2 U D R2 D' F2

Double Y: (no setup needed)
F U' R' B' L2 B R U F'

YJ: (place the correct pair on top left)
F' L U L F D2 F' L' U' L' F

Fifth step: edges
I do this intuitively, you can use things along the lines of M2 U2 M2 U2
(note that it should be easy to generate algs for all cases though, there are not that many)

Final step: edges
again, should be easy to generate algs for it but I use intuitive edge commutators and an occasional H-perm

I hope this information helps anyone else out there interested in AHTA

On the topic of good algsets for FMC, I agree with trangium about wanting to know the 2e2e and 3e algorithms (at least all those at most 8 HTM).

I would like to know what people think of the idea of learning algorithms to optimally (or close to optimally) solving corners once DR has been reached. Up to initial/final U/D moves, and quotienting by rotational and reflection symmetries, there aren't actually that many cases. At least maybe it will help with knowing whether one has good corners or not.

If I had to say how I could guess whether corners are good or not after DR, I'd try and naively get a HTR by putting the corners into their correct orbits under the squares group, and testing if this was actually a HTR (considering corners only, so yes with probability 1/3). If yes, good corners, otherwise no. Any pros out there have any thoughts on this, or insights into how they distinguish between good/bad corners (and if bad corners, do you just throw it away? (oh dear, probably too many questions now))

I would like to know what people think of the idea of learning algorithms to optimally (or close to optimally) solving corners once DR has been reached. Up to initial/final U/D moves, and quotienting by rotational and reflection symmetries, there aren't actually that many cases. At least maybe it will help with knowing whether one has good corners or not.

I think it would be just as useful (and probably easier) to learn how to tell quickly if a DR has easy corners or not. Being close to HTR is a good hint, because if corner are HTR'd then they are at most 3 moves away from being solved (worst case is 4, but you can always cancel one move).

The advantage of following an approach like the one outlined in this document instead of memorizing algs is that it is intuitive, so you can adapt it to solve full HTR or leave only a few edges unsolved.

Hey guys, just began trying FMC a week ago after i saw a couple YouTube videos and I'm really enjoying it. Got a 35 move solve today, and i thought I'd share it with you guys, enjoy!

Scramble: R' U' F R' L' F' L' U F' B' L' U F B R2 L2 D2 F D2 R2 F2 R2 B R' U' F

Skeleton:
2x2x2: F' B D F' L' U B2 (7/7)
2x2x3: (L' F L) (3/10)
F2L-1: D' * D' R2 ** D2 (4/14)
Last slot: D R D' R D' F D F' (7/21)

I was left with a lucky last layer, just a 3 cycle of edges and a 3 cycle of corners, and i found some good insertions i think. Still, last layer is where my main struggles are, so any help is welcome.

*Edge 3 cycle: R' B' F D2 B F' R'
** Corner 3 cycle: R' D2 R U R' D2 R U'

Solution:
F' B D F' L' U B2
D' R' B' F D2 B F' R' D'
R D2 R U R' D2 R U'
D' R D' R D' F D F' L' F' L
Total: 35 moves

Alright guys, that's it from me! Was really excited about breaking sub 40 so i thought I'd share it with you all. As mentioned earlier, any tips are much appreciated!

Hey guys, just began trying FMC a week ago after i saw a couple YouTube videos and I'm really enjoying it. Got a 35 move solve today, and i thought I'd share it with you guys, enjoy!

Scramble: R' U' F R' L' F' L' U F' B' L' U F B R2 L2 D2 F D2 R2 F2 R2 B R' U' F

Skeleton:
2x2x2: F' B D F' L' U B2 (7/7)
2x2x3: (L' F L) (3/10)
F2L-1: D' * D' R2 ** D2 (4/14)
Last slot: D R D' R D' F D F' (7/21)

I was left with a lucky last layer, just a 3 cycle of edges and a 3 cycle of corners, and i found some good insertions i think. Still, last layer is where my main struggles are, so any help is welcome.

*Edge 3 cycle: R' B' F D2 B F' R'
** Corner 3 cycle: R' D2 R U R' D2 R U'

Solution:
F' B D F' L' U B2
D' R' B' F D2 B F' R' D'
R D2 R U R' D2 R U'
D' R D' R D' F D F' L' F' L
Total: 35 moves

Alright guys, that's it from me! Was really excited about breaking sub 40 so i thought I'd share it with you all. As mentioned earlier, any tips are much appreciated!

35 moves is a nice result! I would try to get to only 3 corners unsolved, instead of leaving myself with a last layer. To do this, try many different possibilities on the normal and inverse scramble.

For example, with your 2x2x3, you can do this to get to 3 corners left:

2x2x2: F' B D F' L' U B2 (7/7)
2x2x3: (L' F L) (3/10)
Set up red-blue edge: (D' F D F') (4/14)
F2L-1: (R D2 R2) (3/17)
3c: (R2 B' R B R D R' D' R D) (10-2/25)

The insertion cancels 3 moves, for a total of 25+8-3=30.
Skeleton: F' B D F' L' U B2 D' R' D R D' R' B R' B' D2 R' * F D' F' D L' F' L
* R' F L F' R F L' F' (8-3/30)

It's usually good to orient edges in the beginning or middle of a solve, because misoriented edges get harder and harder to deal with the further along the solve you get. In this case EO after your 2x2x3 only takes 3 moves, and it leads to a nice 29:
2x2x2: F' B D F' L' U B2 (7/7)
2x2x3: (L' F L) (3/10)
EO: (F D' F') (3/13)
Square: (D R') (2/15)
Extend to 1x2x3: (D2 R) (2/17)
Reduce to 2e2c: (D' R D') (3/20)
ZBLL 2e2c: (D' F D' L2 U B' U' L2 D2 F') (10-1/29)

Finally, you should learn Domino Reduction (DR), where all pieces are oriented and the E-slice edges are in the E-slice. It is frequently used by top solvers to get sub-25 results. With DR, corner insertions are worse than normal, but edge insertions are much better. In the solve below, it only took 5 moves to solve 5 edges! Here's a solve using DR:

L R U B // EO (4/4)
L2 U F2 D' L2 D L' // DR (7/11)
B2 D F2 D * R2 U' L2 D L2 D // 5e (10/21)

* D2 L2 R2 U2 ** L2 R2 // 3e (6-3/24)
** F2 R2 D R2 F2 L2 U L2 // Solved (8-6/26)

35 moves is a nice result! I would try to get to only 3 corners unsolved, instead of leaving myself with a last layer. To do this, try many different possibilities on the normal and inverse scramble.

For example, with your 2x2x3, you can do this to get to 3 corners left:

2x2x2: F' B D F' L' U B2 (7/7)
2x2x3: (L' F L) (3/10)
Set up red-blue edge: (D' F D F') (4/14)
F2L-1: (R D2 R2) (3/17)
3c: (R2 B' R B R D R' D' R D) (10-2/25)

The insertion cancels 3 moves, for a total of 25+8-3=30.
Skeleton: F' B D F' L' U B2 D' R' D R D' R' B R' B' D2 R' * F D' F' D L' F' L
* R' F L F' R F L' F' (8-3/30)

It's usually good to orient edges in the beginning or middle of a solve, because misoriented edges get harder and harder to deal with the further along the solve you get. In this case EO after your 2x2x3 only takes 3 moves, and it leads to a nice 29:
2x2x2: F' B D F' L' U B2 (7/7)
2x2x3: (L' F L) (3/10)
EO: (F D' F') (3/13)
Square: (D R') (2/15)
Extend to 1x2x3: (D2 R) (2/17)
Reduce to 2e2c: (D' R D') (3/20)
ZBLL 2e2c: (D' F D' L2 U B' U' L2 D2 F') (10-1/29)

Finally, you should learn Domino Reduction (DR), where all pieces are oriented and the E-slice edges are in the E-slice. It is frequently used by top solvers to get sub-25 results. With DR, corner insertions are worse than normal, but edge insertions are much better. In the solve below, it only took 5 moves to solve 5 edges! Here's a solve using DR:

L R U B // EO (4/4)
L2 U F2 D' L2 D L' // DR (7/11)
B2 D F2 D * R2 U' L2 D L2 D // 5e (10/21)

* D2 L2 R2 U2 ** L2 R2 // 3e (6-3/24)
** F2 R2 D R2 F2 L2 U L2 // Solved (8-6/26)

Wow thanks a lot for the advice, I really appreciate you going through it so thoroughly! I'll definitely keep these tips in mind in the future. Thanks again for the advice!

Nice solve, and congrats for your new records!
I have tried the scramble and I got a 23 (my first good solve in months ahah). I have also found your DR, but not your nice direct finish.

Reconstructions of my 23.67 Spanish NR Mo3 done at OnlyFMC2020

Good DR's that lead to simple and good skeletons with lucky insertions

Spoiler:
R' U' F D' F L2 B' R2 F2 D2 R2 F' D2 F' R' U' R2 D L2 U' L B' D' R' U' F (25)

B D' R//EO (3/3)
F' U'//DR-2e4c (2/5)
B U2 B2 U F2 U'//DR (6/11)
B' R2//sq+line (2/13)
D2 B2 U2//block+good corners (3/16)
B L2 B' F2 R2 F'//AB3E (6/22)

Skeleton: B D' R F' U' B U2 B2 U F2 U' B' R2 D2 * B2 U2 B L2 F2 B' R2 F' 22 to 3e

*D2 R2 U2 F U2 R2 D2 B//3e (8-5/25)

Solve: B D' R F' U' B U2 B2 U F2 U' B' U2 F U2 R2 D2 B' U2 B L2 F2 B' R2 F'

25 moves

Spoiler: R' U' F R2 D B2 F2 D2 U' B2 F2 R2 D' L' R' B L2 R' B' R2 U' B R' U' F (23)

F2 B2 L' F' B//EO+DR-4e4c (5/5)
(R F2 B2 R' D2 L D')//DR (7/12)
(R F2 B2)//123 (3/15)
(R' B2 R')//AB2E2E (3/18)

Skeleton: F2 B2 * L' F' B R B2 R F2 B2 R' D L' D2 R F2 B2 R' 18 to 2e2e

*U2 L2 U2 L2 U2 L2//2e2e (6-1/23)

Solve: F2 B2 U2 L2 U2 L2 U2 L F' B R B2 R F2 B2 R' D L' D2 R F2 B2 R'

23 moves

Spoiler: R' U' F D' R B2 L B2 D2 L' U2 R B L D' R' F U L' U R' D R' U' F (23)

(D' L' D')//EO (3/3)
(B' R)//DR-2e3c (2/5)
(F' D2 B' R B R')//DR (6/11)
(B' U2)//sq (2/13)
(B' D2)//sq+123 (2/15)
(B' R2 B F')//AB3E (4/19)

Skeleton: (D' L' D' B' R F' D2 B' R B R' B' U2 B' D2 * B' R2 B F') 19 to 3e

*(D2 L2 U2 F U2 L2 D2 B)//3e (8-4/23)

Solve: F B' R2 D2 L2 U2 F' U2 L2 B U2 B R B' R' B D2 F R' B D L D

I am looking to get into FMC, right now I average around 35-38 using Petrus, but obviously I am going to need to learn more things if I want to get better. With so many things to learn, where should I start? Thanks in advance!

I am looking to get into FMC, right now I average around 35-38 using Petrus, but obviously I am going to need to learn more things if I want to get better. With so many things to learn, where should I start? Thanks in advance!

I am looking to get into FMC, right now I average around 35-38 using Petrus, but obviously I am going to need to learn more things if I want to get better. With so many things to learn, where should I start? Thanks in advance!

That's a good foundation. Try to shift the EO step to the very beginning of the solve first and of course make sure to not break EO in the following steps.
Then there are two things that are very important to learn:
1. How to finish a solve more efficiently. With Petrus there are many different bad methods what to do after the 223 block / F2L. You will probably want to get to last slot, then just solve all the remaining 5 edges, and then solve the remaining corners with insertions (the "insertions" part is what you should learn).
2. NISS (this can be used together with any other method for fmc)

That alone should be enough to average well under 30 moves. Once you're comfortable with it, you can jump straight into DR (you could in theory do DR directly too, but it's a bit harder to understand although somewhat more powerful).

That's a good foundation. Try to shift the EO step to the very beginning of the solve first and of course make sure to not break EO in the following steps.
Then there are two things that are very important to learn:
1. How to finish a solve more efficiently. With Petrus there are many different bad methods what to do after the 223 block / F2L. You will probably want to get to last slot, then just solve all the remaining 5 edges, and then solve the remaining corners with insertions (the "insertions" part is what you should learn).
2. NISS (this can be used together with any other method for fmc)

That alone should be enough to average well under 30 moves. Once you're comfortable with it, you can jump straight into DR (you could in theory do DR directly too, but it's a bit harder to understand although somewhat more powerful).

Wait how do you do EO in the beginning, isn't there a RUF thing at the the beginning and end of the solve? And aren't you not allowed to do the inverse of those moves?

I want to get decent at FMC. I'm familiar with how EO, blockbuilding and NISS work. I haven't done a full FMC solve yet (lack of time.) What concepts should i learn next?

What's a skeleton, what are insertions? I have questions.

Wait how do you do EO in the beginning, isn't there a RUF thing at the the beginning and end of the solve? And aren't you not allowed to do the inverse of those moves?

The rule is there to prevent people from simply inversing the scramble and write that down as your solution. If you don't do that it's fine.
And I think the misconception here is that you believe if you do those R U F moves EO will be solved but that's not the case. The scrambles are random state which means that every EO case can come up.

A skeleton is an incomplete solution where you solve most, but not all, of the pieces. For example, you could have all but 3 corners solved, and you can fix that with a corner commutator. Sometimes if you try to do the commutator at the end, it takes 8 moves or even more (the maximum is 12); instead, you can track where those three corner pieces to cycle are throughout your skeleton, and find a place where doing the commutator takes only 8 moves, hopefully even cancelling some moves.

The same logic applies to 3e (3 edges), 2e2e (a pair of edge swaps), 2c2e (corner swap + edge swap) skeletons/insertions. Note that you can insert any kind of alg anywhere; the most useful algs are those that affect only a small number of pieces, and many such algs are commutators, but there's nothing wrong with inserting algs that aren't commutators.

A skeleton is an incomplete solution where you solve most, but not all, of the pieces. For example, you could have all but 3 corners solved, and you can fix that with a corner commutator. Sometimes if you try to do the commutator at the end, it takes 8 moves or even more (the maximum is 12); instead, you can track where those three corner pieces to cycle are throughout your skeleton, and find a place where doing the commutator takes only 8 moves, hopefully even cancelling more moves.

The same logic applies to 3e (3 edges), 2e2e (a pair of edge swaps), 2c2e (corner swap + edge swap) skeletons/insertions. Note that you can insert any kind of alg anywhere; the most useful algs are those that affect only a small number of pieces, and most such algs are commutators, but there's nothing wrong with inserting algs that aren't commutators.

It should be fairly obvious once you try it out. (You already know NISS, right?)

An example from one of my solves for the weekly comp (2019-34).

Scramble: R' U' F R' U2 L' D2 U2 L' F2 R B2 L2 B2 D' F L' B2 U' L' U' L2 U' R' U' F

U L' (R B2) // pair and square (4/4)
D B' R' D (D') // 223 (5/9)
(B L' B' L' B2) // another square (5/14)
(B' L B L' B' L2 U' L U B) // all but 3 corners (10-1/23)

Here, I have three corner pieces left to solve. Rewrite the whole skeleton in a linear form:
U L' D B' R' D B' U' L' U @ L2 B L B' L' B' L B L B' D B2 R'

(It's possible to do insertions in the NISS form, but it's harder and you're going to have to write your final solution in linear form anyway.) At the place marked with an @ sign, I can do a corner commutator U R U' L U R' U' L' to solve those three corners, then continue with the rest of the solution. This comm also cancels two moves (the U at the start with the U move before the @; the L' at the end with the L2 move after the @).

U L' D B' R' D B' U' L' U U R U' L U R' U' L' L2 B L B' L' B' L B L B' D B2 R'
= U L' D B' R' D B' U' L' U2 R U' L U R' U' L B L B' L' B' L B L B' D B2 R'

(My solve #2 for that week's comp has an 11-move 4t3e insertion, if you want to see an example of a "unconventional" insertion.)

You just pretend some pieces are solved even if they actually aren't (especially if properly solving them would take a lot of moves), and keep going until you're stuck.

R B F2 U' F' //EO (5/5)
R U' //Reduce to 4c1e (2/7)
(R' D' L2 D) //DR (4/11)

Now one might think that one could setup to a single move (like U) for HTR. But actually we are in the No parity / UD0 / FB1 case of the parity maze. In particular, if we did setup (with double moves only) and then a single move, the parity would be odd, so no HTR.
Instead, we need to first reduce to a different case:

R2 F2 L' //Reduce to Parity / UD1 / FB1

And then we can find an HTR:

B2 L2 U2 L //HTR in 18

Unfortunately the optimal finish for this HTR is 10 moves (U2 F' B' D2 L2 U2 R2 F B R2), so I ended up using my backup solution:

R B (L2 U2 B) //EO
(U' L B2) //321
(U L' D L U2) //322
(D R' D R2) //F2L-1
(D2 R D' R' D' R D2 R' D) //Finish 26