# Updated HTR guide for FMC (Introduction to HTR subsets)



## ottozing (Jan 8, 2023)

Previous guide which you should read if you haven't had a chance to yet as I won't be restating all of it here.

Since posting my last guide, I've learned quite a bit more about HTR and have a more complete understanding of HTR than I did previously. ITT I'll be covering some key information on how HTR subsets work. Throughout this post I'll be assuming the DR is U/D axis, meaning that the top and bottom layers are oriented with the E slice edges oriented and placed.

In total, there are 12 HTR corner subsets ranging from 0-5 quarter turns, and within these 12 subsets you could have a certain number of edges oriented and misoriented. When you account for slice manipulation, you can either have 0 corners misoriented (same as 8 when you insert E), 2 corners (same as 6), or 4 corners. For the purposes of this guide, I'm going to begin by focusing on just HTR-ing the corners since it's the most important thing to know when navigating from unsolved to solved HTR.

*0CXE*

With 0 corner cases, it's either 0qt (corners can be solved with only half turns), 3qt (U R2 U2 F2 U R2 U2 F2 U), or 4qt (U R2 U R2 U2 F2 U R2 U). In my experience, the latter 2 are pretty bad unless the DR is quite short as the HTR solutions are likely to be quite long (6 being best case scenario with NISS or 9 without NISS), so it's not super important you learn how to navigate these cases. Note that for 0qt, this is often really 2qt as you'll need to do one move to get 1qt 4C4E, and then solve that case to get HTR. This theme of going backwards and forwards to get HTR is something you should keep in mind, as it's not uncommon for the optimal solution to get HTR to involve this strategy when it's a case in the 0-2qt range (or even 3qt in some rare awful cases).

*2CXE*

For 2 corner cases, these range from 3-5qt and all follow a structure where the optimal paths from HTR-XCXE to HTR can all be done with only one half turn between quarter turns in a non NISS context. Of all the 4qt cases you can get, 2CXE is by far the best one for this reason. Note that with 3qt 2CXE you can always reach 2qt by simply turning it into 4CXE regardless of how many half turns you insert before doing so (you can also always immediately fix fake HTR "parity" by switching the direction of the first qt move, more on this in the 4CXE section), but with 4qt 2CXE you need to be careful not to accidentally "reduce" to 5qt 2CXE. I would recommend everyone spend some time learning how these non 5qt 2CXE cases work since they're generally not that complicated. 4qt 4CXE also has a funny trick where you can do R U R' B2 R U' R' style tricks to get HTR, though these make it unlikely that you'll find a short re-write to solve the E slice.

*4CXE*

Finally, for 4 corner cases, These range from 1-5qt including two slightly different 2qt possibilities (either U R2 U or U R2 U2 F2 U) which I'll be referring to as 2aqt and 2bqt. Similar to the 0 corner cases, 1qt 4CXE often needs 3qt unless it's specifically 4C4E (even then, you should consider 3qt here). For 2qt, the two case types can be differentiated by whether or not you can do 1qt to give fake HTR/triple DR, or 2qt. Both cases are quite good, though for 2bqt you'll often need to NISS to find short solutions. Note that the fake HTR's you may experience with 4aqt can be avoided by setting up the corners to a position where you need 4 moves to orient corners, and then do a U2. It's good to try fixing this parity at various points of your solution, either at the beginning or after you've made some pairs. For 3qt, this case follows a structure similar to 4qt 0CXE minus two moves either at the start or at the end. These are also not too bad but often require NISS for short solutions. The 4-5qt 4CXE cases suck big time and also aren't super worth it to learn, though I will note the 5qt case follows a similar structure to 5qt 2CXE where the optimal path never needs more than one half turn between quarter turns.

*HTR subsets*

Here's where the fun begins. Once you've identified what type of corners you have, whether through Hyper Parity or simply recognizing the corner case, it's absolutely worthwhile to also take note of how many unsolved edges there are. The reason for this is that different HTR subsets will have different upper and lower bounds in a non NISS context. While it's true that almost any HTR subset can be done efficiently with NISS and some luck, non NISS HTR's have enough benefits in terms of time management and double slice reduction opportunities for these upper and lower bounds to factor into your decision to discard a DR or attempt to finish it.

More research needs to be done into these subsets since I don't believe we know the lower and upper bounds for every subset. This kind of research could also be valuable to get an idea of how often certain approaches lead to the optimal non NISS HTR solution. For example, with 2aqt 4C4E, I've personally noticed that it's pretty common to see either 1qt of progress made followed by a medium length 1qt 4C4E, as well as 1qt of regression made into some 2C2E or 2C4E case. Half turn setups into U R2 F2 U is also a thing, but I'm personally not convinced it's the most common path.

I say this because I want to make it clear that I'm still not an expert on all of these HTR subsets, though I do feel my knowledge on the good sets is enough for the purposes of this initial guide. Below are my current thoughts on which subsets are good/bad within the scope of what I consider to be "good corners", though you can take these thoughts with a grain of salt as some of my claims aren't yet backed by research.

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0qt - Pretty much everything here is good, though I will say 2E and 4E are the best sets & 6E is probably the worst of the 4 (IIRC it has a lower bound of 8 in non NISS scenarios but it's still likely to be quite good with NISS). While I don't know for sure, I suspect 2E might be the best overall as it's always 5-6 moves and has a ton of ways to manipulate the U/D layer leftovers via widening non U/D moves and also inserting moves. The only argument for HTR-4E would be the fact that it has a slightly shorter lower bound of 4 move via U R2 L2 D.

1qt - Also always good, though 4C4E is certainly a lot better than everything else. My current suspicion is that 4C2E is 2nd best within this corner subset.

2aqt - Also always good. I'm guessing here that 4C2E>4C4E>4C.

2bqt - This is where it gets a little interesting. If it's 4C4E then solutions like U R2 U2 F2 U are possible. Otherwise, I've personally noticed that this case tends to be pretty annoying as optimal solutions often show either tricky 4qt non NISS solutions, or complex 2qt NISS solutions.

3qt 2CXE - If it's 2C2E (almost definitely best) or 2C4E (almost definitely 2nd best), it's quite good and may actually be better than non 4C4E 2bqt. Otherwise, it's a bit middling. 2C6E is probably 3rd best as it's the only remaining set where the non NISS lower bound is sub 8 moves. It also has the same NISS lower bound of 4 which you also see with 2C2/4E, but generally speaking it's nowhere near as good as the previous 2 cases as they at least have non NISS 5 movers and lots of 6-7 movers. 2C0E and 2C8E suck big time since the non NISS solutions are awful and you can't make 1qt of progress into 2aqt 4C2E.

3qt 4CXE - I don't really know much about this set other than 4C2/4E have a NISS lower bound of 4, and 4C has 5 for the same reasons outlined in the 2CXE section. As such I suspect the former two cases are a little better but don't know if one is significantly better than the other.

3qt 0CXE - Avoid. I also don't know much about this set other than the NISS lower bound for all sets being 6 and the non NISS lower bound being at least 9 if not more

4qt 2CXE - I did a tiny bit of research into this set and found that 2C, 2C2E, and 2C4E all have at least one non NISS 7 mover which isn't terrible. 2C8E is probably the only one you want to throw out straight away since there's no way to do one move of progress into 3qt 2C2E and it doesn't have great non NISS solutions.

Everything else - Probably avoid

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Here are 3 examples showing what my personal tracing system looks like (looking at the corner case and figuring out the HTR subset based on the number of quarter turns). I'll also include some solutions that I would find either on normal or inverse (I won't cover both for every scramble for the sake of brevity) if there's any non NISS HTR's that I would realistically check. None of the solutions I list will be computer generated, so you're welcome to try using NISSY on these scrambles if you're interested in discussing optimal HTR's as well as optimal solutions & how they may be explainable and findable!

In the future I may complete this set of example's using scrambles that specifically cover the other 9 HTR corner subsets. For now, here's one example each for 2aqt, 3qt 2CXE, and 3qt 4CXE.

~



Spoiler: #1



U2 B2 U2 B2 R2 U2 B2 F2 L2 U2 R2 U' R2 U2 R2 F2 B2 U B2 R2 U' F2 U2 F2 U'

Here I see that the corner case is one that could be solved with an R2 U R2 U' R2 style solution with R2 F2 R2 inserted somewhere to give solved corners. This case will always be in the 2-4qt range based on whether or not you need an AUF before, after, or before and after. Here we don't need any AUF, so this is 2qt (you could also tell this by the fact that it's HTR-4CXE). Here we also see that it's HTR-4C4E, so our HTR subset for this DR is 2aqt 4C4E.

We can also see that the corners are in a good "parity" state on this side of the scramble, so it's possible here to try making 1qt of progress to potentially get a nice 1qt 4C4E case. Here one of the first things I try leads to a promising set of HTR's to try:

R2 D' // Reduce to 1qt 4C4E
F2 (or F2w) L2 * D2 F2 U // HTR

* F2, or B2, or F2 B2

This immediately gives us 2 possible 7 move HTR's to check, as well as 4 8 move HTR's and 2 9 move HTR's. While some of these are on the longer side, they may give a skip which helps illustrate the value in prioritizing non NISS HTR's for time management

Note that due to symmetry, doing an F2 turns this case into its mirror, so we could check a similar set of 8-10 move HTR's with something like F2 R2 U F2 L2 D2 B2 D

We could also try another common idea for 2aqt 4C4E which is doing 4qt instead. In this case, if we do D' we get a promising 3qt 2C4E case since there's a block and a pair. From here I know by heart that it's possible to finish this with L2 U' L2 D' R2 D for another 7 move HTR. There's also a lot of ways to extend this to 9 moves either by inserting half turns before going from 3qt to 2qt, or by inserting slice half turns whenever the case is in an XC4E state with diagonal edges (I'll leave this as a challenge to the reader to find all of these!)

Similar to before, note that we can do an F2 to give ourselves the mirror case and also proceed with this 4qt idea.

Finally, I'll quickly touch on doing half turn setups to U R2 F2 U. Here it doesn't look amazing at first glance, though I see D2 R2 (or R2w) D2 R2 gives us the case while keeping a good parity. This can then be solved with D' B2 (or B2w) R2 D for a set of 8 move HTR's to check.





Spoiler: #2



L2 R2 U2 B2 U2 B2 F2 L2 F2 R2 L2 U' R2 B2 U' F2 U2 F2 L2 R2 U B2 F2 U2 L2

Here we have a very different corner case where the optimal solution follows a structure of R2 U R2 U' R2 U2 B2, which again gives us a range of 2-4qt. This case is a little different to the previous one since we can either get 2aqt (likely with 4CXE "parity" on one side of the scramble), 3qt 2CXE, 3qt 4CXE, or some bad 4qt case which we would likely throw out.

Here I see that we only need an AUF before starting the corner solution and not after, so based on the structure of the solution itself, along with checking the number of corners unoriented, this is 3qt 2C6E. This HTR subset tends to suck and sort of forces you NISS after 1qt of progress to get a decent HTR (HTR in 4 is possible here in theory). However, for instructional purposes I'll show you what I would check without NISS since some 7 movers are possible.

If we do a clockwise U or D turn, we get 2aqt while fixing the "parity" issue at the same time, and in order to get a decent solution I think it's very likely that we'll need to fix this parity straight away (for cases like 2C2E and 2C4E we don't necessarily need to do this, but here I think it's practical to do so). Using the most obvious choice of U, the best I can personally see is trying to set up the 4 move 2aqt 4C4E with B2 F2 R2 followed by 4 variations of the next 5 moves to give a set of 9 move HTR's to check. I'm not convinced I found the optimal solution using this starting move, but after trying a few other ideas I don't see anything better.

Going back to the beginning, Since the misoriented corners are on top of each other, all of the side layers could be turned once (or more, though here we know that won't be very good) before doing our quarter turn of progress. We also know that since the parity fixing move was clockwise, these side turns will change it to counterclockwise.

Trying these 1 by 1, I see that F2 D' gives a 2aqt 4C2E, but I only see a way to solve this case in 7 more moves which is just one measly 9 move HTR, so not really an improvement. R2 U' gives an interesting 2aqt 4C4E which is solvable in 7 more moves but at least has 4 variations to check (hint, B2 is the first move). L2 U' here seems to give basically the same case as F2 D' which we already discussed, and B2 U' seems to give us what we already had using our initial U move which we know isn't very good.

While I'm not going to go into the NISS HTR stuff since it's a pain, I will say that if I was going to keep this DR (unlikely), I would try switching after F2 D and L2 U as these give us a subset which is 3 optimal, and I would also switch after U since it's only the cost of one move. I may also try switching after one of the other setups if there was a cancellation outside of the DR that makes it only cost me one move. This process I just went through is pretty much the exact same as what I would do on the other side of the scramble. Since this process is pretty time intensive and doesn't give me much else to check every time I NISS, this DR would have to be very short relative to my other DR's for me to seriously consider it.





Spoiler: #3



U' B2 U' B2 R2 B2 U2 R2 F2 R2 U2 L2 F2 L2 R2 F2 U2 B2 U' B2 U2 R2 B2 R2 L2

This is one of those corner cases that isn't very easy to recognize without Hyper Parity. You could try learning these diag/diag orientation cases with some sort of brute force, but I think this set is where Hyper Parity is likely better. Alas, I am trash and don't know it so I'll just show you how I'd figure it out

After doing an R2, I see that we have something that resembles the 2x2 ortega JJ PBL, so this is immediately either 2bqt or 3qt 4CXE depending on whether or not we need a quarter turn AUF afterwards. Here we do, so we know that it's 3qt 4C2E. As such, the solution structure will follow some sort of U R2 U R2 U2 F2 U structure. Another way to think about this case is as a 2aqt case where you're 1-2 moves away from giving yourself fake HTR on one side of the scramble. If you're going to check a DR with this type of corners, you want to check this "almost fake HTR" side first due to the possibility of short NISS HTR's. Here we just so happen to be on that side of the scramble which helps.

One annoying thing about this case is that you need to be careful with how you do the half turns before making a qt of progress, and by paying attention to the possible corner solutions I already know I'll need at least two half turns. I would almost certainly throw this DR out after checking the other side once just to see if there's a simple way to do 1qt of progress followed by the U R2 U2 F2 U 4C4E 2bqt case.


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