# 3x3 PBL Theoretical Method (3BL) (CFOP Variant)



## Duncan Kimmett (Jan 2, 2023)

So, I invented a theoretical CFOP variant known as "3BL". It consists of a few steps:
1.) Solve the cross ANY WAY YOU WANT (so it doesn't have to align the colors, just connect the cross-colored part of the edges to the center)
2.) Solve the rest of the cube up until PLL
3.) Do a PBL algorithm to solve both cross and PLL at the same time

I also have a document that shows the method more in-depth like showing the obvious many conns, and obvious or less well-known pros. I will have this document linked below. Anyone who opens the document can give comments/suggestions that I will consider.

Document: https://docs.google.com/document/d/1X36fviqTDq4BV1ZY1udoPfikENzc4tLdZ-LfknV8QcU/edit


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## ProStar (Jan 2, 2023)

So you're saving a couple moves on the cross to add what is probably 5+ moves at the end of the solve (since it replaces PLL)? It seems like a massive inrease in alg count for a time drop of basically 0, plus unless the algs are super optimized it'll actually be slower than just doing the cross and ending with PLL. It'll also make recognition annoying since you'll have to remember at the end of the solve what the cross case was, which you would've solved at the beginning


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## Duncan Kimmett (Jan 2, 2023)

ProStar said:


> So you're saving a couple moves on the cross to add what is probably 5+ moves at the end of the solve (since it replaces PLL)? It seems like a massive inrease in alg count for a time drop of basically 0, plus unless the algs are super optimized it'll actually be slower than just doing the cross and ending with PLL. It'll also make recognition annoying since you'll have to remember at the end of the solve what the cross case was, which you would've solved at the beginning


I agree with all of these statements and actually already have thought of most of them, I just didn’t care about this being a serious method and wanted to share it with people. But thank you for the healthy criticism, I have heard some new things I have not already heard of and I think that is a big win.

Thanks!


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## Thom S. (Jan 2, 2023)

1. This has been proposed before. You can't know that, but that also excludes you from being the inventor.
2. If you're not serious about the method, why tell us about it?
3. Layer 1 and Layer 3 are pnly interchangable through R2 or M2, the former needing a lot of moves to bring both corners back to the cross. Now imagine that with L, F, R, B / M, S all at the same time.


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## Duncan Kimmett (Jan 2, 2023)

Thom S. said:


> 1. This has been proposed before. You can't know that, but that also excludes you from being the inventor.
> 2. If you're not serious about the method, why tell us about it?


First of all, I could guess this has been proposed before, but I shouldn’t be named the inventor even though I called myself it, so that is my blame. Second, I was bored. I was 15 on 3x3 and I normally average 14 so I took a break, got bored and decided to post this. Lastly, I already said these algs would be long but also, you are definitely right because my only probable way for this is most likely doing PLL then solving a U perm or something, like the stupid idea of 3 look 3bl. I can confirm this method is absolutely illogical and is not worth using, I just wanted to get it out there because it is an interesting idea (at least I think) and although people have proposed it before, but I went a little more in depth with offering my pros and conns and letting people suggest. Sorry if you see that as bragging, I just believe it is something that needs its own document and thread because the last one did not use a document. The one that I have been referencing is a proposition from MrIndianTeen, the thread being named ”Can PBL Be Used In 3x3??.” Thank you!


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## Cubinginatree (Jan 2, 2023)

Duncan Kimmett said:


> So, I invented a theoretical CFOP variant known as "3BL". It consists of a few steps:
> 1.) Solve the cross ANY WAY YOU WANT (so it doesn't have to align the colors, just connect the cross-colored part of the edges to the center)
> 2.) Solve the rest of the cube up until PLL
> 3.) Do a PBL algorithm to solve both cross and PLL at the same time
> ...


I thought I made this method up.. Oh well this is cool.


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## Cubinginatree (Jan 2, 2023)

other people can edit your document by the way


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## Duncan Kimmett (Jan 2, 2023)

Cubinginatree said:


> other people can edit your document by the way


I set it to suggest ( I care about them not being able to delete stuff)


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## Duncan Kimmett (Jan 2, 2023)

Cubinginatree said:


> I thought I made this method up.. Oh well this is cool.


I'm sure many people have thought of this, I actually mentioned someone who made the exact same method and posted it.


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## Duncan Kimmett (Jan 2, 2023)

I have started making the algorithms. I am doing this for fun, keep in mind, so I don't mean for people to learn them.


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## Duncan Kimmett (Jan 2, 2023)

Oh by the way, the math was wrong, it is 252 algorithms not 336.


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## Filipe Teixeira (Jan 2, 2023)

ewww


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## Jorian Meeuse (Jan 2, 2023)

Isn't it 168?
Bottom can be Ua, Ub, W, Oa, Ob, H, Z, Adj, Diag
Top can be all 21 pll's
H bottom is nonsense, you can just do D2 after cross
This leaves 8 x 21 = 168 cases excluding normal PLLs


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## Duncan Kimmett (Jan 3, 2023)

Jorian Meeuse said:


> Isn't it 168?
> Bottom can be Ua, Ub, W, Oa, Ob, H, Z, Adj, Diag
> Top can be all 21 pll's
> H bottom is nonsense, you can just do D2 after cross
> This leaves 8 x 21 = 168 cases excluding normal PLLs


Thank you SO MUCH! I literally have been trying to figure it out for ages. Thank you! Mind if I use your explanation to explain the math?


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## abunickabhi (Jan 3, 2023)

Duncan Kimmett said:


> Oh by the way, the math was wrong, it is 252 algorithms not 336.


168 is a lot of cases to be honest.

Do you think the method is worth it?


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## cubenerd74 (Jan 3, 2023)

abunickabhi said:


> 168 is a lot of cases to be honest.
> 
> Do you think the method is worth it?


Well at least it's fewer cases than 5-style


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## Thom S. (Jan 3, 2023)

We should more talk about quality of algorithms.
ZBLL is a big algset, too, but 5/6 of the cases are good.


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## Duncan Kimmett (Jan 3, 2023)

As the creator, I think we could cut down a ton of algs actually because H perm cross and O perm cross are just a few moves away from solved cross. Personally, I probably won't learn it, but maybe I can cut it down enough.


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## Duncan Kimmett (Jan 3, 2023)

Duncan Kimmett said:


> As the creator, I think we could cut down a ton of algs actually because H perm cross and O perm cross are just a few moves away from solved cross. Personally, I probably won't learn it, but maybe I can cut it down enough.


If it becomes 100 or less or just a lil' more I think I will learn it.


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## Jorian Meeuse (Jan 3, 2023)

Duncan Kimmett said:


> As the creator, I think we could cut down a ton of algs actually because H perm cross and O perm cross are just a few moves away from solved cross. Personally, I probably won't learn it, but maybe I can cut it down enough.


Oh yeah, true, O perms are obviously also just one move away after doing cross, I didn't even realise that lol. That makes it 6 x 21 = 126 cases excluding normal pll's


Duncan Kimmett said:


> Mind if I use your explanation to explain the math?


Yeah sure, go ahead.


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## Duncan Kimmett (Jan 3, 2023)

Jorian Meeuse said:


> Oh yeah, true, O perms are obviously also just one move away after doing cross, I didn't even realise that lol. That makes it 6 x 21 = 126 cases excluding normal pll's
> 
> Yeah sure, go ahead.


Awesome!


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## Duncan Kimmett (Jan 3, 2023)

Jorian Meeuse said:


> Isn't it 168?
> Bottom can be Ua, Ub, W, Oa, Ob, H, Z, Adj, Diag
> Top can be all 21 pll's
> H bottom is nonsense, you can just do D2 after cross
> This leaves 8 x 21 = 168 cases excluding normal PLLs


Wait. When bottom isn't an alg top isn't a legit PLL. How do I figure these algs out?!?!?!


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## Duncan Kimmett (Jan 3, 2023)

Duncan Kimmett said:


> Wait. When bottom isn't an alg top isn't a legit PLL. How do I figure these algs out?!?!?!


I have a shaky plan which is do a ton of solves with that cross on bottom then see the PLLs


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## Jorian Meeuse (Jan 3, 2023)

Duncan Kimmett said:


> Wait. When bottom isn't an alg top isn't a legit PLL. How do I figure these algs out?!?!?!


What do you mean 'when bottom isn't an alg'? The line and W cases?


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## Duncan Kimmett (Jan 3, 2023)

Jorian Meeuse said:


> What do you mean 'when bottom isn't an alg'? The line and W cases?


So, W and adjacent edges and opposite edges give combined PLL cases. I can give a screenshot.


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## Jorian Meeuse (Jan 3, 2023)

Duncan Kimmett said:


> So, W and adjacent edges and opposite edges give combined PLL cases. I can give a screenshot.


Yeah that's just like PLL parity on 4x4. Since the bottom has an odd number of swaps, the top must too. The easiest case is M2 U2 M2.


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## Duncan Kimmett (Jan 3, 2023)

Jorian Meeuse said:


> Yeah that's just like PLL parity on 4x4. Since the bottom has an odd number of swaps, the top must too. The easiest case is M2 U2 M2.


Exactly! But I need help on figuring out how to find all the cases. How can I know if I didn't find a case?


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## Jorian Meeuse (Jan 3, 2023)

Duncan Kimmett said:


> Exactly! But I need help on figuring out how to find all the cases. How can I know if I didn't find a case?


Well if you have 126 cases total, you have all algs. If you want to know which cases there are possible on the top, just imagine taking each regular pll and doing a single swap of edges, e.g. a diagonal swap, to each of them. That means there are 22 possible PLL cases. Wait I'm just now realizing that changes the math. 21 cases on the top for Ua, Ub, Z, 22 cases for Adj, Diag, W, gives 21 x 3 + 22 x 3 = 129 cases.


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## Duncan Kimmett (Jan 3, 2023)

Jorian Meeuse said:


> Well if you have 126 cases total, you have all algs. If you want to know which cases there are possible on the top, just imagine taking each regular pll and doing a single swap of edges, e.g. a diagonal swap, to each of them. That means there are 22 possible PLL cases. Wait I'm just now realizing that changes the math. 21 cases on the top for Ua, Ub, Z, 22 cases for Adj, Diag, W, gives 21 x 3 + 22 x 3 = 129 cases.


Thank you so much! You make me feel like an idiot!


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## Jorian Meeuse (Jan 3, 2023)

Duncan Kimmett said:


> Thank you so much! You make me feel like an idiot!


Nope you're not 
If you want to check if you have every case, this video is about 4x4 PLL parity, but it covers every possible case so you can maybe use it as a checklist.


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## Duncan Kimmett (Jan 3, 2023)

OMG THIS IS AMAZING!!!!!


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## Duncan Kimmett (Jan 3, 2023)

also imma get back to making algs now. I'll check this every once in while


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## Duncan Kimmett (Jan 3, 2023)

Just finished all the U Perm Cross Cases!


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## Silky (Jan 3, 2023)

PLL parity only occurs if you have O, W, Adj, or Opp on the D face. To avoid PLL "Parity" you can preform M2 U2 M2 ( this transforms your O, W, Adj, Opp into a EPLL ). All EP cases have already been generated and can be found here, both with and without "Parity".

Also there are 109 non-Parity cases ( including PLL ) and 110 Parity cases. 219 in total.


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## Cubinginatree (Jan 3, 2023)

How do you generate the algs?


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## Duncan Kimmett (Jan 3, 2023)

Cubinginatree said:


> How do you generate the algs?


If you're asking me, I use a thing called "cube explorer" which finds optimal algs for every case by the click of a button.


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## Duncan Kimmett (Jan 3, 2023)

Silky said:


> PLL parity only occurs if you have O, W, Adj, or Opp on the D face. To avoid PLL "Parity" you can preform M2 U2 M2 ( this transforms your O, W, Adj, Opp into a EPLL ). All EP cases have already been generated and can be found here, both with and without "Parity".
> 
> Also there are 109 non-Parity cases ( including PLL ) and 110 Parity cases. 219 in total.


I think that might be too many algs for my taste. Imma just stake with our current strat. Do people agree with that?


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## Duncan Kimmett (Jan 4, 2023)

I'm just gonna assume Ob-Perm cycles to the left while Oa perm cycles to the right?


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## Imsoosm (Jan 4, 2023)

I would suggest you to use the Batch Solver as cube explorer's optimal algorithms usually have very bad ergonomics. Batch Solver can generate a set of algorithms at a time, and can also check for the alg with highest ergonomics. 

Here is the link to the original thread, you might need to watch the tutorial:








The Batch Solver - Generate large algorithm sets automatically!


The Batch Solver Batch Solver Tutorial by OreKehStrah Introduction Before the batch solver, generating algorithms involved inputting each and every case into Cube Explorer, running the search for that one case, testing out the generated algorithms to determine the best one for that case, then...




www.speedsolving.com


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## hyn (Jan 4, 2023)

Imsoosm said:


> I would suggest you to use the Batch Solver as cube explorer's optimal algorithms usually have very bad ergonomics. Batch Solver can generate a set of algorithms at a time, and can also check for the alg with highest ergonomics.
> 
> Here is the link to the original thread, you might need to watch the tutorial:
> 
> ...


ur signature says 3x3 Qiyi MS

didn't u have a gan 12?


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## Duncan Kimmett (Jan 4, 2023)

Imsoosm said:


> I would suggest you to use the Batch Solver as cube explorer's optimal algorithms usually have very bad ergonomics. Batch Solver can generate a set of algorithms at a time, and can also check for the alg with highest ergonomics.
> 
> Here is the link to the original thread, you might need to watch the tutorial:
> 
> ...


Thank you very much! I'll try it out! Do you think it would be worth re-doing an absolute ton of algs just for this, or should I just continue with it?


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## Duncan Kimmett (Jan 4, 2023)

Nvmnd. It has something that cube explorer offers that it doesn't. I'm working with 4x4 parity algs, and since cube explorer has a painting feature, it is very easy to implement them. This does not do that.


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## Imsoosm (Jan 4, 2023)

hyn said:


> ur signature says 3x3 Qiyi MS
> 
> didn't u have a gan 12?


I did, but I liked the Qiyi MS over the gan 12 lol


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## CornerTwisted (Jan 4, 2023)

cubenerd74 said:


> Well at least it's fewer cases than 5-style


I can't get over this.


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## Jorian Meeuse (Jan 4, 2023)

Silky said:


> Also there are 109 non-Parity cases ( including PLL ) and 110 Parity cases. 219 in total.


How did you get those numbers? 

Btw good point, you did include the pll skip on top which I didn't, but I feel like you also included H and O cases
Including the pll skip on top but not H and O cases it would give
6 x 22 = 132 cases excluding normal PLL
That is actually a big con of this method, you never get a PLL skip lol


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## Duncan Kimmett (Jan 4, 2023)

Just finished the Adj. cross edges cases, all 22.


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## Duncan Kimmett (Jan 4, 2023)

Jorian Meeuse said:


> How did you get those numbers?
> 
> Btw good point, you did include the pll skip on top which I didn't, but I feel like you also included H and O cases
> Including the pll skip on top but not H and O cases it would give
> ...


Oh that is a massive con. I think it could be worth it since almost all algs where you would have had PLL skip are extremely easy


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## Duncan Kimmett (Jan 4, 2023)

Jorian Meeuse said:


> How did you get those numbers?
> 
> Btw good point, you did include the pll skip on top which I didn't, but I feel like you also included H and O cases
> Including the pll skip on top but not H and O cases it would give
> ...


Also PLL skip is just f2 then your U perm and f2 (f2 (Ub/Ua perm) f2), and then Z perm is just double rotate.


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## Duncan Kimmett (Jan 5, 2023)

Just finished the Opp. Cross edges cases!


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## Duncan Kimmett (Jan 5, 2023)

Jorian Meeuse said:


> How did you get those numbers?
> 
> Btw good point, you did include the pll skip on top which I didn't, but I feel like you also included H and O cases
> Including the pll skip on top but not H and O cases it would give
> ...


Are you considering that there is no such thing as PLL skip on Adj., Opp., W,


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## Jorian Meeuse (Jan 5, 2023)

Duncan Kimmett said:


> Are you considering that there is no such thing as PLL skip on Adj., Opp., W,


Wdym? Of course there isn't


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## Duncan Kimmett (Jan 5, 2023)

Jorian Meeuse said:


> Wdym? Of course there isn't


Sorry, I got confuzzled, I guess it was because I wrote this at 10 pm. I do not understand what I was thinking.


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## Silky (Jan 5, 2023)

Jorian Meeuse said:


> How did you get those numbers?
> 
> Btw good point, you did include the pll skip on top which I didn't, but I feel like you also included H and O cases
> Including the pll skip on top but not H and O cases it would give
> ...



Just plugged it into Batch, lol. But your breakdown is correct:

21 PLLs x 4 EPLLs = 84
PLL = 21
EPLL on D = 4
22 Parity PLLs x 5 Parity EPLLs = 110
Total : 219

So 219 - 22 -22 -21 -21 = 133 excluding PLL, H, Oa, Ob.
Edit: -1 for H + PLL Skip. 132 Total.


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## Duncan Kimmett (Jan 5, 2023)

So It is 133 Algs instead of 129 Algs?


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## Silky (Jan 5, 2023)

Duncan Kimmett said:


> So It is 133 Algs instead of 129 Algs?



I gave the entire breakdown above. You tell me, idk what you are or are not generating.


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## Jorian Meeuse (Jan 5, 2023)

Silky said:


> Just plugged it into Batch, lol. But your breakdown is correct:
> 
> 21 PLLs x 4 EPLLs = 84
> PLL = 21
> ...


2 mistakes here: for the O cases, you should also subtract 22. This makes for a final 131. But then, at the start, you said PLL = 21, and you proceed to subtract 22 for PLL at the end, making the total of 131 1 short. The actual number is 132. Just 6 x 22.


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## hyn (Jan 6, 2023)

U can reduce the EPLL on D cases to just Solved, Z and one of the Uperms using just D moves
Adj and W can D or D' to either Uperm, Opp is a D or D' away from a Z perm, and the Os have been accounted for
So it'll just be 63+2 = 65
D or D' just does 3 edge swaps so parity on bottom is easily avoidable


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## Silky (Jan 6, 2023)

Jorian Meeuse said:


> 2 mistakes here: for the O cases, you should also subtract 22. This makes for a final 131. But then, at the start, you said PLL = 21, and you proceed to subtract 22 for PLL at the end, making the total of 131 1 short. The actual number is 132. Just 6 x 22.



I didn't list the cases in order. All the numbers are correct tho.



hyn said:


> U can reduce the EPLL on D cases to just Solved, Z and one of the Uperms using just D moves
> Adj and W can D or D' to either Uperm, Opp is a D or D' away from a Z perm, and the Os have been accounted for
> So it'll just be 63+2 = 65
> D or D' just does 3 edge swaps so parity on bottom is easily avoidable



Can you show an example of this? This doesn't sound right to me.


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## hyn (Jan 6, 2023)

Just swap DF and DR, then do a D


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## Duncan Kimmett (Jan 6, 2023)

Silky said:


> I gave the entire breakdown above. You tell me, idk what you are or are not generating.


I am not gonna generate the algs for u perm cross or z perm cross because I think you could probably just rotate twice for z perm and for U perm: f2 (Ub perm) f2, or f2 (UA perm) f2.


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## Duncan Kimmett (Jan 6, 2023)

hyn said:


> U can reduce the EPLL on D cases to just Solved, Z and one of the Uperms using just D moves
> Adj and W can D or D' to either Uperm, Opp is a D or D' away from a Z perm, and the Os have been accounted for
> So it'll just be 63+2 = 65
> D or D' just does 3 edge swaps so parity on bottom is easily avoidable


I thought of that, but one of the best pros is the x-cross ability, and having to align a piece makes x-crosses harder. Idk, I think that I should include that as a thing to mention in the doc to let people know if they ever want to learn algs, only 65 are really NEEDED. Your opinion?


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## Duncan Kimmett (Jan 6, 2023)

hyn said:


> U can reduce the EPLL on D cases to just Solved, Z and one of the Uperms using just D moves
> Adj and W can D or D' to either Uperm, Opp is a D or D' away from a Z perm, and the Os have been accounted for
> So it'll just be 63+2 = 65
> D or D' just does 3 edge swaps so parity on bottom is easily avoidable


Also, if you don’t do W perm perm with aligning the edge perfectly, then you just get Adj. case. Just something to think about.


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## Jorian Meeuse (Jan 6, 2023)

Silky said:


> I didn't list the cases in order. All the numbers are correct tho.
> 
> 
> 
> Can you show an example of this? This doesn't sound right to me.


No, it is 132 cases. Everytime you have 21 in your breakdown, it should be 22.


What hyn suggestes is indeed correct. He means that if you have parity on the cross, you can do a D or D' to get no parity.


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## Cuber987 (Jan 6, 2023)

This is a really good theoretical idea! I'd love to try it out if we ever find all the algs.


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## Silky (Jan 6, 2023)

Jorian Meeuse said:


> No, it is 132 cases. Everytime you have 21 in your breakdown, it should be 22.



We agree, I said 132. Reread the post.


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## Jorian Meeuse (Jan 6, 2023)

Silky said:


> We agree, I said 132. Reread the post.


Oops sorry i didn't catch that


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## Duncan Kimmett (Jan 6, 2023)

Cuber987 said:


> This is a really good theoretical idea! I'd love to try it out if we ever find all the algs.


Oh, I’m finishing all the algs today.


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## Duncan Kimmett (Jan 6, 2023)

I’m at 115 algs of 132 (More like 129) so the algs will be finished today.


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## Duncan Kimmett (Jan 6, 2023)

I am very glad to announce I have finished every single algorithm!


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## Jorian Meeuse (Jan 6, 2023)

Duncan Kimmett said:


> I am very glad to announce I have finished every single algorithm!


Well that's ambiguous.


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## Duncan Kimmett (Jan 6, 2023)

Jorian Meeuse said:


> Well that's ambiguous.


Check the doc. I have Pll skips, and all the algs for W Perm cross, Z Perm cross, Ua Perm cross, Ub Perm cross, Adj. cross, and opp. cross!


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## Jorian Meeuse (Jan 7, 2023)

Duncan Kimmett said:


> Check the doc. I have Pll skips, and all the algs for W Perm cross, Z Perm cross, Ua Perm cross, Ub Perm cross, Adj. cross, and opp. cross!


Awesome! 
(With ambiguous I meant that 'finishing' algs can also mean 'killing' algs)


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## Duncan Kimmett (Jan 7, 2023)

Jorian Meeuse said:


> Awesome!
> (With ambiguous I meant that 'finishing' algs can also mean 'killing' algs)


Ah! That’s my bad!


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## Duncan Kimmett (Thursday at 3:02 AM)

Could you do EFLL (Edge Flip Last Layer)? Have a flipped cross edge, and then you do an alg to fix the swapped edges on both cross and PLL. Math: 1 edge flipped on PLL, 4 different edge positions, so (21 PLLs) (4 Positions) = 84. 84 x 4 because you can have 1 flipped edge, 2 flipped edges, 3 flipped edges, and 4 flipped edges. My math is probably very wrong, because 2 flipped cross edges and 4 flipped cross edges is 21 algs each, so I’ll just fix the math in the final math, meaning 21 + 21 = 42 and then 84 x 2 = 168. 168 + 42 = 210. Any thoughts?


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## Silky (Thursday at 12:44 PM)

Duncan Kimmett said:


> Could you do EFLL (Edge Flip Last Layer)? Have a flipped cross edge, and then you do an alg to fix the swapped edges on both cross and PLL. Math: 1 edge flipped on PLL, 4 different edge positions, so (21 PLLs) (4 Positions) = 84. 84 x 4 because you can have 1 flipped edge, 2 flipped edges, 3 flipped edges, and 4 flipped edges. My math is probably very wrong, because 2 flipped cross edges and 4 flipped cross edges is 21 algs each, so I’ll just fix the math in the final math, meaning 21 + 21 = 42 and then 84 x 2 = 168. 168 + 42 = 210. Any thoughts?



You cannot have 3 flipped edge pieces


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## Duncan Kimmett (Thursday at 11:05 PM)

Silky said:


> You cannot have 3 flipped edge pieces


ah, thanks


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## Duncan Kimmett (Thursday at 11:06 PM)

math is still correct


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## Duncan Kimmett (Thursday at 11:24 PM)

Algs are too long


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