DR-Xs

By Janne Lehtimäki (@tseitsei89 on discord)

Contents:

What is this?...........................................................................................................................0

Main method…..……………………………………………………………………………………..1

Slice EO..……………………………………………………………………………………………1.1

rzp-Xs...……………………………………………………………………………………………...1.2

DR-2s………………………………………………………………………………………………...1.3

DR-1s………………………………………………………………………………………………...1.4

DR……….……………………………………………………………………………………………1.5

Direct DR-2s -> DR…………………………………………………………………1.5.1

HTR…………………………………………………………………………………………………...1.6

Finish………………………………………………………………………………………………….1.7

When is this good?......…….…………….………………………………………………………...2

Movecount comparison…………………………………………………………………………….2.1

Findability comparison………..……………………………………………………………………2.2

My recommendations………………………………………………………………………………2.3

Example solves……………………………………………………………………………………….3

What else to try for DR-1s -> finish…………………….…………………………………………4

Variation spamming………………………………………………………………………………...4.1

HTR - fake slice (HTR-f1s).....................................................................................................4.2

Random tips…………………………,……………………………………………………………….5

-Xs prediction………………………………………………………………………………………..5.1

EO altering rzp………………………………………………………………………………………5.2

Slice insertion to change -Xs………………………………………………………………………5.3

DR-2s 8 flipped edges………………………………………………………………………………6

How to get DR-2s 8fe………………………………………………………………………………6.1

HTR-2s………………………………………………………………………………………………6.2

Finish……………………………………………………………..…………………………………6.3

Movecount and findability comparison……………………….………………………………….6.4

My thoughts…………………………………………………………………………………………6.5

What is this?

This is a way to find FMC solutions where we leverage many concepts from DR/HTR but we leave some EO fixing until the very end of the solve. This allows us to find solutions normal DR/HTR can not find. This method will most likely NOT be good on all solves but on some scrambles it will reach better results than normal DR/HTR meta. I will talk more about when to use this later in chapter 2 of this doc.

This method is mostly not developed by me but rather a community effort from at least Jay, Rodney Kinney, and discord users: @trangium, @count quackula, @KrzychXyz and most likely many more I don’t remember or even know about.

Also if you notice something wrong or weird or missing in this doc feel free to contact me on discord (@tseitsei89).

NOTE: in order to use this method you already need to be quite fluent with normal DR/HTR since this method uses much of the same concepts but with harder recognition at many points.

1 Main method:

1.1 Slice EO

Instead of doing full EO to begin with we only care about EO of pieces that belong to a single slice. And we can either orient all of these pieces OR misorient all of these pieces. As long as all of the 4 slice edges have the same orientation it is good. On most scrambles there will be 0-1 move solutions for this since we have many different EOs we can try. We can choose any slice (S, E, M) and look at orientation on any axis (U/D, R/L, F/B) and we can orient or misorient.

I do not recommend looking at > 1 move slice EOs in a real solve.

Example:

FMCL S9W3A2:
R' U' F U R F B' U2 F' L2 U B2 L' F' R2 D2 F' R2 F B' L2 U' R' U' F

We have a slice EO skip here on normal for:

F/B E-slice (all oriented)

F/B S-slice (all oriented

and on inverse

F/B E-slice (all oriented)

F/B M-slice (all oriented)

U/D M-slice (all misoriented)

And 1 move slice EO on R/L E-slice with: L (misoriented)

Normally there are not this many skips but there are often 1-2 on a scramble and some 1 movers.

NOTE: NISS does NOT preserve partial EO.

But you can also look for NISS slice EOs by trying to do 1 move to get all pieces currently in a given slice to have the same orientation. This will give slice EO when you NISS.

1.2 rzp-Xs

This step is simple once we get used to ignoring some pieces since it is so similar to normal DR. We just completely ignore the flipped edges and centers and try to look for rzp like normal.

We want to look at corner orientation in such a way that our (mis)oriented slice edges will be between the DR layers. For example if we have E-slice oriented for F/B axis you need to look at U/D stickers. And you can look at them in both U/D and R/L faces.

Example:

FMCL S9W3A2:
R' U' F U R F B' U2 F' L2 U B2 L' F' R2 D2 F' R2 F B' L2 U' R' U' F

E-slice is already oriented on F/B so we just look at rzp right away. On U/D here we have 4c6e and doing R gives us 4c4e in 1 move. On R/L axis we have 7c4e and no short way of getting rzp. We can do R L U for 4c4e. This is also a valid rzp since we just ignore the center color. But that is already 3 moves and just like normal DR we want rzp to mostly be 0-2 moves.

On inverse however we have a great case. E-slice is still oriented on F/B and doing R gives us 3c2e in 1 move!

NOTE: NISS does NOT preserve rzp-Xs!

But while NISS doesn’t necessarily preserve rzp it DOES preserve the DR minus case. The problem is that you might have some DR-slice edges flipped on the other side. But, if you look at edges currently in DR-slice for your rzp and they all have the same orientation, the rzp is preserved when you NISS. In our example both R and (R) rzps have all edges in E-slice oriented so we can still use NISS on those!

1.3 DR-2s

This step is also simple once we get used to ignoring unimportant pieces. We just ignore flipped edges and centers and do rzp->DR like normal.

Also the freedom of ignoring non-DRslice EO also allows us to use some additional DR triggers that change EO of those pieces. Here is a full list of additional sub5 triggers for most commonly used rzps:

F R F / F’ R F //3c2e
R’ F R F’ //sledge for 4c2e

B U’ B’ R //4c4e more easily remember as Fw R’ Fw’ R = inverse sledge with widened F moves

M F // 4c4e

M U2 F // 4c4e

F R U' L // 7c8e

R F2 R U R L // 7c8e (you can get rid of the R L so it's really a 4 mover)

R U2 R U R L // 7c8e

R U(w) L2 D R L // 7c8e

Example:

FMCL S9W3A2:
R' U' F U R F B' U2 F' L2 U B2 L' F' R2 D2 F' R2 F B' L2 U' R' U' F

In previous step we found the promising 3c2e rzp on inverse so we use that to continue:

(R’) //rzp-Xs 3c2e

(D2) //setup to trigger

(L U L) //DR-2s in 5 moves

The reason we call this DR-2s is because we need 2 slice moves to fix flipped edges and centers. Here centers are already solved but we have 2 flipped edges. So our first slice must unsolve centers and leave exactly 4 flipped edges. Then the second slice can simultaneously put centers back to correct places and flip all 4 edges.

Sometimes you can also get DR-3s cases but then you can always go to DR-2s in 0 moves by inserting a slice move that cancels with your last DR move. Sometimes you skip this step and get DR-1s instantly. That is obviously great.

NOTE: from this step onwards you can use NISS again

1.4 DR-1s

Kind of a new but super simple step. We just want to do a slice move that leaves us with:

1. centers 1 move from solved
2. exactly 4 flipped edges.

After that real DR can be achieved with just a single slice move.

Often this slice can be done directly after DR at least on one side of the scramble. Sometimes you might need 1 extra move before slicing

Example:

FMCL S9W3A2:
R' U' F U R F B' U2 F' L2 U B2 L' F' R2 D2 F' R2 F B' L2 U' R' U' F

Previously we found:

(R’) //rzp-Xs 3c2e

(D2) //setup to trigger

(L U L) //DR-2s in 5 moves

Now on inverse we have 2 flipped edges and solved centers so we only need to get these edges in different slices and we get 4e while doing either M or S.

Here we obviously decide to do M’ since it cancels with the last move of DR-2s. So we have:

(L’ R) //2-2 = DR-1s in 5.

1.5 DR

Next we want to fix the slice to get to real DR.

We do this by setting all flipped edges in the slice where centers are offset and doing the slice to get to real DR. Don’t worry too much about the length of our DR at this point. It will end with U R L’ type moves so it will be much better than move count suggests. And we can also try multiple different setups quickly. And we plan to slice the R L’ out later anyway.

Note that it is very important to try many different slice fix move sequences and pick the ones with best subsets. Usually I would recommend trying all possible sub7 move fixes (so up to 4 setup moves +2 moves for slice itself).

Example:

FMCL S9W3A2:
R' U' F U R F B' U2 F' L2 U B2 L' F' R2 D2 F' R2 F B' L2 U' R' U' F

Previously we found:

(R’) //rzp-Xs 3c2e

(D2) //setup to trigger

(L U L) //DR-2s in 5 moves

(L’ R) //4a2-1s in 5 moves

The shortest possible setups for edges are 5 moves both on normal: D’ B2 U R L’  and on inverse: (F’ R2 F R L’)

But both of these give 2c4 subset. We can add one move on normal to change that:

F2 D’ B2 U R L’ //4b2  in 11 moves

        1.5.1 Direct DR-2s ->DR

        Once we get more familiar with the method we can combine steps 1.4 and 1.5 and see

the shortest possible slice fixes directly from DR-2s instead of just doing the shortest

possible fix to DR-1s first.

Example:

R' U' F R2 U' F L' D F2 R D F' R F' R2 L2 D2 F R2 L2 F R2 F B R' U' F

(U F R2 B L2 U2 B') // dr-2s in 7

 Now if we strictly follow the method described above we get something like:

(U D) //shortest possible reduction to dr-1s in 9

        (R2 F2 D2 B R L) //dr in 15 so obviously not great… (Also many other variations

        give DR in 15)

        But, we can instead do:

        (R2 U D) //dr-1s with extra move that sets edges up for short slice fix

        (B R L) //dr in 13 so 2 moves better!

1.6 HTR

After reaching DR we want to switch to the other side of the scramble and spam linear HTRs from there

Example:

FMCL S9W3A2:
R' U' F U R F B' U2 F' L2 U B2 L' F' R2 D2 F' R2 F B' L2 U' R' U' F

Previously we found:

(R’) //rzp-Xs 3c2e

(D2) //setup to trigger

(L U L) //DR-2s in 5 moves

(L’ R) //4a2-1s in 5 moves

F2 D’ B2 U R L’ //4b2  in 11 moves

Now we NISS and can just start spamming HTRs like normal.

Shortest possible HTR for us here is:

(R2 B2  U2 R2 B’ D2 B) // 7-1 = HTR in 17 moves

1.7 Finish

After HTR you don’t really want to focus much on finding a direct leave slice. If you happen to see one go ahead but the main point is to find a double slice that leaves the “normal” DR slice and the slice we did the slice move on to reach DR.

After that you want to solve the “wrong” slice while slicing out the R L’ we did to reach DR.

Example:

FMCL S9W3A2:
R' U' F U R F B' U2 F' L2 U B2 L' F' R2 D2 F' R2 F B' L2 U' R' U' F

Previously we found:

(R’) //rzp-Xs 3c2e

(D2) //setup to trigger

(L U L) //DR-2s in 5 moves

(L’ R) //4a2-1s in 5 moves

F2 D’ B2 U [R L’ //4b2 in 11

(R2 B2  U2 R2 B’ D2 B’) // 7-1 = HTR in 17 moves

(U2 [L2 D2 F2) //double slice 4/21

[]= U2 R L F2 U2 //slice solved by slicing out the “DR fixing slice” LS in -2/19

TIP: If there is no short solution for the slice, you can also try to solve it to a case where centers are 180 degrees off.

That can most of the time be fixed in +0 (and always in +1) by inserting any slices in HTR-1 solution that result in the slice turning 180 degrees from original.

Imagine that we had this HTR line instead in our example:

(R2 B2  U2 R2 F’ R2 F’) // 7-1 = HTR in 17 moves

And the corresponding LS solution is:

(D2 R L B2 D2 R L’) //last R L’ cancels with the last slice before DR.

Now U/D centers are still wrong in addition to normal LS. But now if we go to our HTR-1s solution and widen either both F’ moves OR the F2 the centers will be fixed since those moves rotate S slice centers by 180 while only affecting LS edges.

Now for solving the final LS you can use any method you would normally use. But there is one more additional trick you can try.

There are actually 2 places where you can try solving the LS. The HTR-1s is an obvious one but you can also do it during the edges setup stage!

So in our example we have a solution up to LS:

{F2 D' B2 U' F2} R' L' [U2 B D2 B R2 U2 B2] R U' L' D2 R

In [] we have our HTR-1s solution and best we can do there is [] = U2 F L2 F’ B2 R2 U2 B2 // +1.

But in {} is our edge setup solution and there we can simply widen both U/D moves to get

{} = F2 Dw’ B2 Uw’ F2 //+0.

You can also try combination of both but you obviously have to return centers to solved on the same sequence. So no widening 1 qt from {} and another from [].

But for example {F2 Dw’ B2 Uw’ F2} and [U2 Bw D2 Bw R2 U2 Bw2] is a valid attempt since both {} and [] individually return their respective slice centers back to solved.

Congrats, you now know how to find and solve DR-Xs!

Next we discuss when you should use it.

2 When is this good?

We only have 1 hour to do an FMC attempt so we cannot possibly check everything. So when should we use this new technique? To answer this we need to first look at average move counts of both DR -> solved and DR-Xs -> solved.

2.1 Movecount comparison

This method is still developing so this may change in the future, but currently I think it is not humanly viable to find (near) optimal solutions from DR-2s since those often seem to delay the first slice fix to seemingly random point of the solution which makes it too hard to find for now. But from comparing my practice solves with NISSY output (low sample size but best we have right now) it seems that from DR-1s the above method will often be able to find optimal. And if optimal is unfindable we can then find optimal+1 most of the time.

So in this chapter we compare the optimal movecounts from DR -> solved against DR-1s -> solved for commonly used subsets. Then we can figure out how short DR-1s we need to find for it to give the same average length of overall solution than let’s say 10 or 11 move DR. Thanks to @trangium for generating the stats.

2.2 Findability comparison

So based on the above stats an average finish from DR-1s is about 2.2 - 2.3 moves longer than from DR. However we have to take into account that we can’t always find that optimal from DR-1s. We can’t always find that optimal from DR either but it is more common for optimal to be findable from DR than from DR-1s. So we need to give some value to that as well.

Luckily @trangium was able to generate some stats on how often these optimal solutions are findable with the method described above.

We have not yet generated them for all subsets but we have data for 4b2 and 2c4 cases.

Here a solution to DR-1s is considered findable if

1. it uses < 5 move slice fix (< 7 moves to real DR)
2. it uses < 9 moves to linear HTR from opposite side of slice fix (< 10 for 2c4 since checking 9 move HTRs for 2c4 is quite reasonable)

And a solution for normal DR is considered findable if it uses any < 9 (again < 10 for 2c4) move HTR. Linear from either side and NISS HTR are all allowed for normal DR.

Above table has findability stats for 4b2-1s. So it seems optimal is findable 51% of the time and optimal +1 is findable 83% of the time. So compared to optimal we lose 1 move 32% of the time (83-51=32) and we lose 2 moves 17% of the time (100-83). So on average we lose 0.32*1 + 0.17*2 = 0.66 moves compared to optimal.

Here are the same stats for normal 4b2 DR. Similar math we did above shows us that from this we lose on average 0.255*1 + 0.09*2 = 0.435 moves compared to optimal.

Below are the same stats for 2c4:

So for 2c4-1s we lose on average 0.22*1 + 0.35*2 = 0.92 moves.

And for standard 2c4 DR we lose on average 0.218*1 + 0.147*2 = 0.512 moves.

So for 4b2 we lose 0.66-0.435 = 0.225 moves more from DR-1s than we do from DR.

And for 2c4 it is 0.92-0.512 = 0.408 moves more from DR-1s than from DR.

So let’s take about the average of that and call DR-1s 0.3 moves worse findability wise than DR. This number might get more accurate if/when stats get generated for more subsets.

2.3 My recommendations

So our absolute optimal from DR-1s is something like 2.2-2.3 moves worse than from DR. And, as discussed above, findability adds another 0.3 moves to that. So in total DR-1s seems to be about 2.5-2.6 moves worse on average than DR.

Meaning that 8 move DR-1s is on average better than an 11 move DR but worse than 10 move DR.

So what should we do in an actual solve?
My recommendation based on the above is this:

1. Check all sub5 EOs normally
2. Before starting to spam 5 move EOs check sliceEO skips and maybe 1 move sliceEOs since there is a decent chance for 6-7 move DR-Xs there and some of those might be DR-1s as well. There are normally not that many sliceEOs in 0-1 moves so it won’t take much time.
3. If you still don’t have enough then start spamming 5 move EOs.

3 Example solves

We already did one example solve during the explanation of the method. I will put it below in its entirety once again and then we will do a few more examples.

Example 1:

FMCL S9W3A2:
R' U' F U R F B' U2 F' L2 U B2 L' F' R2 D2 F' R2 F B' L2 U' R' U' F

(.) //F/B E-slice EO skip 0/0

(R’) //rzp-Xs 3c2e 1/1

(D2) //setup to trigger 1/2

(L U L) //DR-2s 3/5

(L’ R) //4a2-1s 2-2/5

F2 D’^ B2 U^ [R L’ //4b2 4/9

(R2 B2  U2 R2 B’ D2 B’) //HTR 7-1/17

(U2 [L2 D2 F2) //double slice 6/21

[]= U2 R L F2 U2 //LS -2/19

^=w //+0/19

Example 2:

FMCL S9W3A3:

R' U' F R' D' F U2 R U2 F2 L2 D2 U2 B2 R D' L' B' R D R F2 R' U' F

First we see that there are slice EO skips on:

R/L E-slice

U/D E-slice

(R/L E-slice)

We actually find 3 DR candidates here:

. //R/L E-slice and also rzp 2c2e on U/D

U’ B’ D’ B2 U B //2c4-3s via well known 2c2e trigger or you can see it as 2 move rzp to 4c4e 6/6

And as we know whenever we get DR-3s we can get to DR-2s in +0 by inserting a slice that cancels:

B’ F // 2c4-2s 2-2/6

Then we have 4 flipped edges so we need to split them 2-2 between the slices so we can remain at 4 flipped edges and bring centers 1 slice away from solved. Fortunately we already have that so we can immediately do:

U D //2c4-1s 2/8

We also know that on inverse there will not be immediate slice fix since edges are in 3-1 split between slices.

At this point we notice that on normal our shortest possible slice fix is 5 moves B2 L2 B R L’ (that goes to 4b5 so not good) and quick NISS trace tells us that on inverse we have a promising 4 move DR (R2 D R L’) that actually goes to 2c3^[a][b].

And we also find these DR-1s:

. //U/D E-slice and also rzp 7c8e on F/B

B2 U2 B R' B R //2c3-3s 6/6

R’ L //2c3-2s 2-2/6

Now we can do U D on normal for 2c3-1s but no 4 move DRs on either side.

Or on inverse we can do (F B) that also has 4 move DR on normal U2 F U D’ that goes to 2c4.

And the variation of the same DR:

B2 U2 B’ L’ B R //4b5-2s and no +0 to -1s so we probably just discard that outright…

So the most promising one we found seems to be the first DR-1s because of the 4 move DR that goes to a good subset. So we continue with that:

U’ B’ D’ B2 U F //2c4-2s 6/6

U D //2c4-1s 2/8

(R2 D [R L’) //2c3 2/12

Then we just spam HTR from normal side:

B^ U2 L2 B’ U2 B’^ //HTR 6/18

L2 F2 //two more pairs 2/20

U2 [R2 F2 //double slice 5/23

[]= R L U2 //slice -1/22

^=w //+0/22

Example 3:

333.fm daily 26.1.2026 (Rodneys DR-Xs and my finish)

R' U' F B2 D F2 U' B2 D U2 R2 F2 D L2 F' L U B' R' D U2 R' F2 D' R' U' F

We have slice EO skips on:

F/B E-slice

R/L E

R/L S

R/L M

U/D M

(U/D E)

(F/B S)

(R/L S)

(U/D S)

(U/D M)

Wow that is a lot!

(.) //U/D E 4c2e rzp 0/0

(L B2 R’ F’ R2 F) //2c4-3s 6/6

(F’ B) //2c4-2s 2-2/6

NISS trace shows us that on normal we have immediate slice to -1s.

In fact we can do either R L or F B. From these options F B gives 3 move DR that goes to 2c3 4e!

F B //2c3 4e -1s 2/8

L [F B’//2c3 4e 3/11

And again here we NISS and start spamming HTR-1s:

(B2 R2 U’ F2 B2 L2 U’ F2 U’) //HTR 9-1/19

(L2 U2 [R2) //double slice 3/22

[]= F B’ D2 //slice +0/22

4 What else to try from DR-1s -> finish?

Obviously this DR-1s -> finish method is not even close to perfect and thus not all optimal solutions will be findable with that. It is just something that

1. works relatively well quite often.
2. can always be done more or less systematically.
3. can be done in  a reasonable amount of time.

So the most reasonable thing in a real solve is often, just like in normal DR, abandon the DR-1s if it doesn’t give a short finish quite quickly.

But sometimes you want to really squeeze the one single DR or DR-1s, for example if that is the only thing you found or super short etc.

So there are few other things you can try.

4.1 Variation spamming

So normally we do edge setup, switch and spam HTR-1s. The reason we do this is that for edge setup+slice fix there are always SO many variations and extensions that you need to check if you do the slice fix right after the edge setup to get real DR and then try HTR from there.

But some solutions are unfortunately only findable by spamming HTRs from as many of the variations as possible.

You can try linear HTRs from the same side where you reach real DR and NISS-HTRs, although NISS-HTRs are probably way too slow to check realistically. Our normal method won’t find those. But checking linear HTRs from the other side is exactly what we check for all variations and extensions simultaneously with our main method so don’t waste time on those.

This has the best chance of working if you have short linear HTRs available and not that short HTRs on the other side where you normally would want to do HTR-1s.

Obvious downside is that you have so many variations that it takes a long time to check them.

Now I don’t happen to have a solve at hand where this would actually find the optimal but example how to do it would be like this:

FMCL S9W3A2:
R' U' F U R F B' U2 F' L2 U B2 L' F' R2 D2 F' R2 F B' L2 U' R' U' F

(.) //F/B E-slice EO skip 0/0

(R’) //rzp-Xs 3c2e 1/1

(D2) //setup to trigger 1/2

(L U L) //DR-2s 3/5

(L’ R) //4a2-1s 2-2/5

F2 D’ B2 U //4b2-1s with flipped edges set up to correct slice 4/9

Then we just do real DR here:

R L //4b2 2e in 11 with TONS of variations and +1 extensions.

Now just try linear HTRs+finish from here like you would from normal DR.

4.2 HTR - fake slice (HTR-f1s)

Sometimes it is also possible that optimal is found by going to what I call HTR - fake slice.

This is a cube state where corners and all oriented edges are in HTR, but in addition you have 4 flipped edges that are split 2-2 between different slices and centers on one slice are 90 degrees off. Also one final requirement is that 2 flipped edges are in the slice they actually belong to and 2 are NOT. This makes it so that you can solve all other pieces and then insert U M U type insertions somewhere post HTR that fixes centers and flipped edges.

This usually seems to happen mostly on low qt subsets where you have really short HTR-f1s solution(s) available. So I would probably only consider it on 4a1 and 4b2 if you have short reductions to 4a1.

The downside is that it is very hard (at least for me) to see solutions from HTR to that state where we only need the U M U insertion. But maybe people with better overall cubing intuition will see them better. Also, even if you see that, the insertion might not have a good spot. Also also the recognition for HTR-f1s is kinda horrible.

But here is an example for an optimal finish like that. The DR-1s is 10 moves and not really realistic to look at especially since there is an 8 move one from the same setup and different slicing but we only want to showcase the finish here really.

Random practise scramble from Rodney:

F2 U' L2 B2 D2 R2 B2 U2 L' B2 L U2 B2 R F D F' R2 F R'

D’ F’ L F2 U2 D2 L’ U // 4b2-2s (8)

F B’ // 4b2-1s (10)

U //4a1-1s (11)

(B2 U2 L) //HTR-f1s (14)

As you can see we now meet all requirements for this state:

1. centers on one slice are 90 degrees off
2. flipped edges are split 2-2 between slices
3. 2 of the flipped edges are in their correct slice (at FL and DR spots) and 2 others are not (at UR and BR spots)

(B2 L2 ^ U2 R2 D2 z) // (19) all but 4 centers and 4 flipped edges. I have no tips on how to see this. Just build blocks and git gud I guess.

^=L F Fw’ L’ //solved 4-1 (22)

And we need to remember to add z at the end for NISS to work if we use the wide move.

5 Random tips

Here are just some random tips I didn’t think were important enough for the main part.

1)  -Xs prediction

After sliceEO you can already sometimes see that it is impossible to get a direct DR-1s on a given axis. Namely if your centers are 3 slice moves away from solved you can’t get -1s. And if your centers are 1 slice off and you do NOT have 4 bad edges you can’t have -1s.

Note that you might still be able to do a slice insertion described in tip 3) to get -1s.

2) EO altering rzp

After slice EO we can also try rzp moves that would normally break EO if we happen to have a situation where none of our slice edges are (or all are) in that layer.

For example if we have F/B E-slice EO we can try F and F’ moves if none of the E edges are in F layer.

3) Slice insertion to change -Xs

Sometimes we can insert a slice move in the DR line (possibly with cancellations) in order to manipulate whether we get -1s or -2s DR. Again the slice needs to be inserted in a spot where it doesn’t affect our slice EO edges.

This seems to be relatively rare however and mostly happens on rzps where most DR slice edges are already in their slice, like 3c2e or 4c2e.

For example if we have F/B E-slice EO (all edges oriented) again with the DR line of:

F2 ^ U’ R U2 R // this gives DR-2s since centers are solved but some edges are flipped.

Now we can insert ^ = S in +1 move since none of the E-slice edges are in S-slice at that point.

This will put centers 1 slice away from solved and can give us DR-1s if we also happen to have 4 flipped edges after that.

6 DR-2s 8 flipped edges

WIP

As discussed above, finding (close to) optimal solutions from DR-2s is often not humanly viable. But there seems to be one exception to that where the solutions are significantly better and more human findable. That is a case where all of the non-DR slice edges are flipped. This allows us to go straight to HTR-2s and solve from there. These cube states seem to have only 1 move worse optimal solutions than normal DRs. Again thanks to @trangium for the stats. More on this in the movecount and findability comparison chapters.

Note that this method is even more in the early stages of development than the DR-1s method described above. So we don’t really have a proper idea on how good this method is or is not. But next I will describe the steps for this method and give my own personal thoughts about its potential, but keep in mind that I have next to no experience using this method in real solves so my opinions might change as I get a better understanding about this stuff.

 6.1 How to get to DR-2s 8fe

There are several ways to get to this cube state. I will go through all of the ones I know about and shortly discuss about how viable I think that method is to use in real solves.

1. You can just happen to stumble on it while trying to find DR-1s like described in the main part of this tutorial. That is obviously viable in a real solve since you just find these as a byproduct when searching for DR-1s. But it seems quite rare to happen “on accident”.

2. Whenever you find a normal DR that ends in moves of the form U R L you can just do D instead of U R L and get DR-2s 8fe that is 2 moves shorter than the real DR. Also very viable since you find these as a byproduct of searching normal DRs. Also note that you can get this if your EO ends in U R L and then you find the DR on the other side (this only works for 1 of the possible 2 DR axis but worth keeping in mind).

        Example:
        R' U' F B L2 F' L2 B F2 U2 R2 F' R2 D' R' B' D F L U' F2 L2 R' U' F

R2 D (U')

F' (L F B) // 7 2c3 6e
        but now instead of DR line we do:
        F’ ( R ) // 5 moves to 2c3 -2s 8fe
        
        Here is another one where the U R L part is on the EO:
        R' U' F R2 L F' L D R2 U' L F' B' R2 B2 U F2 U2 D' F2 D F2 R' U' F

(F2 L F' B)

L2 U2 D L' R2 U // 10 4a1 on L/R, but instead of first line we do
        (F2 R) // 8 moves to 4a1 -2s 8fe

3. You can do normal EO and then solve the DR on “wrong axis” and you get this state. So let’s say we do EO on F/B axis. Then we could do R/L colored “DR” on U/D axis and that would give us DR-2s 8fe. (Or vice versa and do U/D colored DR on R/L axis.) This method also seems quite viable since we are checking normal EOs for standard rzps anyway so simultaneously checking for these rzps shouldn’t take much extra time.

        Example:

        Daily 2026-02-02

R' U' F L2 F D2 B L2 B' L2 F' L2 D2 U2 R F' D U' L' F L' B' R2 U2 R' U' F

(D F R) // EO

(D) // "rzp" 4c4e, we have 4 oriented U/D corners on F/B, and two E-slice edges in S

(F U' F2 L2 D) // 9 moves to 2c3 -2s 8 fe

4. You can force the issue by doing “EO” in such a way that DR slice edges are oriented and all other edges are not and then solve to DR-2s 8fe directly from there. This doesn’t seem as viable since you would have to spend time finding all these “EOs” and checking them as well.

        Example:
        Weekly 2026-02 A1

R' U' F U2 B' L2 B D2 R2 F2 L2 B' F2 U' L' D' B' F2 R' D2 U B' L' R' U' F

(F2 U' B') //orient all E edges and misorient all other edges

(R D2 R' L2 U2 L U') // 10 moves to 4c3 -2s 8fe

6.2 HTR-2s

Okay now we know how to find these DR-2s 8fe solutions. Next we will learn how to solve them.

This next step is quite simple if you already know HTR. Although the recognition can feel quite hard at first.
What we do is ignore centers completely and imagine all flipped edges are just oriented in their current spots. Then we just solve HTR like normal. But instead of real HTR this gives us a cube state that is solvable by doing HTR moves and exactly 2 slice moves to fix centers and orient the 8 edges that are currently flipped.

Example:
R' U' F B L2 F' L2 B F2 U2 R2 F' R2 D' R' B' D F L U' F2 L2 R' U' F

R2 D (U')

F' (L F B) // 7 2c3 6e
but now instead of DR line we do:
F’ ( R ) // 5 moves to 2c3 -2s 8fe
(D2 B') R2 D2 R' B2 L' // HTR-2s in 12 moves

And another one:

Daily 2026-02-02

R' U' F L2 F D2 B L2 B' L2 F' L2 D2 U2 R F' D U' L' F L' B' R2 U2 R' U' F

(D F R) // EO

(D) // "DR"-4c4e, we have 4 oriented U/D corners on F/B, and two E-slice edges in S

(F U' F2 L2 D) // 9 "2c3"

(B R2 F') B2 R2 U' // HTR-2s in 15 moves.

6.3 Finish

Next part is in my opinion the biggest question mark of this method. We want to solve leave slice from here. But the question is how realistic is it for humans to see these solutions quickly. If this turns out to be possible most of the time this method has a lot of promise.

Example:
R' U' F B L2 F' L2 B F2 U2 R2 F' R2 D' R' B' D F L U' F2 L2 R' U' F

R2 D (U')

F' (L F B) // 7 2c3 6e
but now instead of DR line we do:
F’ ( R ) // 5 moves to 2c3 -2s 8fe
(D2 B') R2 D2 R' B2 L' // HTR-2s in 12 moves

F B R’ L’ //16 to slice and +2 to solve.

In this case this is obviously easy to see but usually the finish is not that short.

And another one:

Daily 2026-02-02

R' U' F L2 F D2 B L2 B' L2 F' L2 D2 U2 R F' D U' L' F L' B' R2 U2 R' U' F

(D F R) // EO

(D) // "DR"-4c4e, we have 4 oriented U/D corners on F/B, and two E-slice edges in S

(F U' F2 L2 D) // 9 "2c3"

(B R2 F') B2 R2 U' // HTR-2s in 15 moves.

(D U B F' R2 F2) // 21 to slice +1 to solve.

Note that just like in the DR-1s we can also leave slice AND have the opposite centers swapped. We can do a similar center fix we do there. Example of that here:

R' U' F R2 L F' L D R2 U' L F' B' R2 B2 U F2 U2 D' F2 D F2 R' U' F

L2 U2 D L' R2 U (F2 R') // 8 moves to 4a1 -2s 8fe

(U2 F’) // 10 to HTR-2s

(^ U2 R L F’ B L2 U2) // 17-1 to 2e2e and center fix

^= S2 //2e2e +1/17

So after “DR” we have:

(U2 F B2 D2 * R L F’ B L2 U2)

*= D2 B2 U2 F2 U2 B2 //+2/19

Leave slice can always be solved normally. If you use NISS HTR-2s you will have 2 spots where you can insert slice moves just like we have when solving DR-1s. If we do linear HTR-2s we only have 1 sequence to try like in normal DR/HTR solve.

6.4 Movecount and findability comparison.

First of all @trangium has once again generated some stats to compare optimal solutions from DR and DR-2s 8fe!

So it seems that optimal from DR-2s 8fe is around 1 move worse than from real DR on average. This seems quite promising.

However I am not entirely convinced yet that solutions from HTR-2s -> leave slice are always realistically and quickly findable for humans. Sometimes they definitely are. And I don’t have enough experience with this to approximate how often they are or are not findable at this point.

But at least it should be worth it to check the corresponding DR-2s 8fe if you get a DR that ends in U R L kind of moves. That seems to give optimal quite often if you can just find it. Here are some stats about that for 4b2 subset:
-  | 13.72 (21.3%)

D  | 13.89 (38.0%)

U  | 13.38 (23.7%)

UD | 14.37 (17.0%)

How to read those is like this:
If we have a 4b2 DR ending in U R L moves then we have 38% chance to find optimal by doing D move to DR-2s 8fe instead and solving HTR-2s in < 9 moves and then finish. We have 23.7% of finding optimal by doing linear < 9 HTR from inverse and then solving for double slice and slice out the R L part. And we have 17% chance that optimal can be found by doing either of the methods above.

That leaves us with only 21.3% chance that we do NOT find optimal if we try both of these.

Thanks for reading! That is all I have to say for now. Any feedback and improvements, both on this doc and especially on the method, are most welcome!