# Simultaneous EO and dedge pairing for ZZ on 4x4

#### Cride5

Simultaneous EO and edge pairing for ZZ on 4x4

After experimenting with this method for a while, I'm starting to think it may just be viable option for using ZZ on 4x4.

The basic idea is to orient edges while pairing them. This is achieved by using a varient of Robert Yau's edge pairing method, and solving edges using only l,L,r,R and U. To quote from my initial post on the idea:

It works like so:

The move used to pair these edges would be l U' R U l'

Looking at what happens during these moves, the blue-reds have their orientation preserved, the yellow-orange in B is flipped, and connected with the other yellow-orange, and the white-green in F is flipped and connected with the other white-green. So each time the pairing move is made, 2 edges are flipped. The idea is to set up the dedge pairings so that unoriented edges are flipped to oriented as far as possible.

Just to add claricication. When searching for edge pairs, one of three possible scenarios can arise.
(1) Both edges are oriented. In this case you would pair them in the blue-red positions.
(2) One is oriented and the other is flipped. Here you have two options, they can be paired in either the orange-yellow, or green-white positions.
(3) Both are flipped. Here you use the alternative algorithm: l U' R U l2 U R' U' l, to pair and flip them at the same time. This alg may also pair up to two extra case 2 edges.

When I first looked into this approach I thought it probably wouldn't be practical to do two dedges at a time, but on further investigation it looks like 2 oriented dedges can be created around 50% of the time. Once a pair has been selected there are typically two different adjoining edges. For each adjoining edge the probability that it may be paired with another, is the probability that it, and its matching edge are flipped correctly for pairing. This is 0.5 * 0.5 = 0.25. Since there are usually two edges to choose from, the probability that at least one of them will be flipped correctly becomes: 0.25 + 0.25 = 0.5
These are the probabilities assuming a solver blindly chooses the first edge pair, without considering the state of its adjoining edges. After practising this for a while, I'm starting to get a better feel for good edges pairs to choose, and how to position them to provide the maximum probability of orienting two.

Just to check this I did some slow solves using this edge pairing method, and recorded the number of dedges created during a single pairing move. The results (along with final edge flip count) are as follows:

No. edges paired:
2, 1, 1, 2, 2, 1, 3 = all oriented:
2, 2, 1, 2, 1, 3 = 1 flipped (EO match on final 3)
2, 2, 1, 2, 1, 3 = 1 flipped
2, 2, 1, 1, 1, 3, 2 = 1 flipped
1, 1, 2, 1, 2, 2, 2 = 2 flipped
1, 2, 2, 2, 2, 2 = 2 flipped
1, 2, 1, 2, 1, 2, 3 = all oriented (EO match on final 3)
1, 2, 2, 1, 1, 2, 3 = all oriented
1, 2, 1, 2, 2, 2, 2 = 1 flipped
1, 2, 2, 1, 1, 2, 3 = all oriented

Avg number of pairing moves: (7+6+6+7+7+6+7+7+7+7)/10 = 6.7
Avg number of edges paired per move: (12+11+11+12+11+11+12+12+12+12)/(7+6+6+7+7+6+7+7+7+7) = 1.73
Avg number of flipped dedges after pairing: (1+1+1+2+1)/10 = 0.6

So that's the edge pairing method. To fit it into a ZZ-style solve the steps would be:

1. Solve Centres
2. EOpair DF/DB and place
3. EOpair 10 remaining dedges
4. Flip final bad dedges (inc single flip parity)
5. ZZF2L
6. COLL
7. EPLL with parity (1 alg)

* Good ergonomics during edge pairing as no F/B moves are needed.
* Shorter edge flip parity alg can be used, since only permutation of the line needs to be preserved.
* Finishing with COLL has the advantage that there are relatively few algs to learn to solve parity and EPLL at the same time.
* Solving the line at the start of edge pairing fills some of the worst locations for finding edges.
* Final 3x3 phase is just RUL blockbuilding followed by LL, no pause for inspection of EOLine or Cross.

* More moves during dedge pairing.
* Good ability to see EO during lookahead is required, but this should be pretty natural for ZZ users already.
* Lucky dedges after solving centres may initially may be flipped, these lucky dedges will need to be oriented at the end. The fewer moves required in pairing does mitigate this somewhat though.
* In some cases, flipped dedges may accidentally be created but this should be avoidable with good lookahead.

Algorithms:

Standard Pairing (use mirror for left)
l U' R U l'
r' U' R2 U r

Pair 2x unoriented edges (place edges in B and FR positions)
l U' R U l2 U R' U' l

Final 6 edges where all orientations match:
F {pairing alg} F'

Final 4 edges - oriented (dedge in UL flipped)
r' F R2 U' R' F' U r
r' F U' R' F' R2 U r

Final 4 edges - flipped (dedge in UL flipped)
r' U' F R' U F' r F U2 R' F'
r' F U' R F' U r R' B L' U2 B'

Flip single edge (in DR):
y x r U2 r' U2 r' D2 r D2 r' B2 r B2 r'

Example Solve:

Scramble: D' r2 D u' r2 u2 d' f' D' B F' r2 D' f2 R2 D' U' B2 D2 l2 b2 u2 F' r2 U' B2 f2 U2 F' l2 b U' F u' d r D2 d' F2 L

L+R Centres: u2 r' R' L2 f U l U' l' z' (9)
Finish Centres: r' U r U l' U l2 U2 l' x U2 r2 U2 r2 (13/22)
First line dedge (+dedge): L U D' L' U r' U L' U' r (10/32)
Second line dedge: R U' R r' U' R2 U r (8/40)
Place line: R' L' D' (3/43)
1x dedge: U2 R' l U' R U l' (7/50)
2x dedges: U2 r' U' R2 U r (6/56)
3x dedges: L U L' r' U L' U' r (8/64)
Final 3 dedges: L' U' l U' R U l' U R' U' l U' R U l' (15/79)
Finish EO: B L' B' (3/82)
ZZF2L: R2 L' U2 L2 R2 U2 R U' R' U R' U' R U' R' U R U2 L' U L U L' (23/105)
OCLL: y2 R U2 R' U' R U' R' (7/112)
PLL: y x R D' R U2 R' D R U2 R2 x' U' (10/122)

... for future reference, I think I'm going to call this Z4

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#### CubeNoobie

##### Member
Hit me If I am wrong, but I see a huge disadvantage....
It seems like the solver has to look ahead for the orientation instead of the next pairs...

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#### Cride5

Hit me If I am wrong, but I see a huge disadvantage....
It seems like the solver has to look ahead for the orientation instead of the next pairs...
Yup, this is mentioned in the disadvantages section, but personally I don't find it to be too much of a problem. You have to be aware of the edges' orientations relative to each other during pairing anyway, otherwise they won't fit together. Using this just means that a solver needs to be aware of how the orientations will change after pairing.

#### mariano.aquino

can you already do a speed solve with Z4?
looks interesting
i use K4, but like to explore different methods. and i would already know Yau part =P

#### mariano.aquino

can you already do a speed solve with Z4?
looks interesting
i use K4, but like to explore different methods. and i would already know Yau part =P

#### Moops

##### Member
Interesting method. May have potential to be really fast. I'm going to play with it for little while and compare it to my redux solving.

#### Petro Leum

##### Member
hey cride! sorry for diggign the thread out... do you actually use your method? i am experimenting t ouse ZZ efficiently for the 4x4 now for a few days, and i also have a yau-similar approach - however mine does not orient and pair the dedges simultaneously. i would love to talk with you about 4x4 methods ;D

Oh, and, do you have the alg for the one parity/Epll case? thanks

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#### Pyjam

Like you Petro Leum.

I do some Yau-ZZ :
1) 2 centers
2) 3 pairs including futur DF+DB + any other pair, all must be oriented.
4) last 4 centers
5) 4th pair, any, must be oriented, to complete the false cross.
6) last 8 pairs : 3-1-1-3, any orientation
7) orient the last 8 pairs (not that difficult)
8) orientation parity
9) finish in ZZ.

It's the fastest for me, although I'm not very fast.

Edit: Sorry, I didn't see your last msg was from 2012, not 2013.
Also, in France, 07-03-2012 means March, 7th.

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#### Berkmann18

##### Member
Huge bump.
Is there anyone who tried doing Z4 the Hoya way other than me ?
1) F4Ce
2) EOLine
3) 4 dedges to form two 1x2x2 blocks in the back
4) L2Ce
5) L6E+EO (+EO parity)
6) ZZ finish with preferably COLL/EPLL as Conrad suggested or ZBLL.

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#### Petro Leum

##### Member
Huge bump.
Is there anyone who tried doing Z4 the Hoya way other than me ?
1) F4Ce
2) EOLine
3) 4 dedges to form two 1x2x2 blocks in the back
4) L2Ce
5) L6E+EO (+EO parity)
6) ZZ finish with preferably as Conrad suggested or ZBLL.
at this point i dont even know how hoya works, but i might have a look at it.

#### Berkmann18

##### Member
at this point i dont even know how hoya works, but i might have a look at it.
Hoya is:
1) F4Ce
2) Cross dedges (so F4E)
3) L2Ce
4) L8E
5) 3x3 stage

#### Y2k1

##### Member
Another major bump, where are the epll plus parity algs? I really like this method, thanks

#### Lazy Einstein

##### Alg Ninja
Huge bump.
Is there anyone who tried doing Z4 the Hoya way other than me ?
1) F4Ce
2) EOLine
3) 4 dedges to form two 1x2x2 blocks in the back
4) L2Ce
5) L6E+EO (+EO parity)
6) ZZ finish with preferably COLL/EPLL as Conrad suggested or ZBLL.
I am looking into this before I start doing big cube ZBLS.

I like your steps. I would probably do two dedges in D during EOline to transition easier.

#### Berkmann18

##### Member
I am looking into this before I start doing big cube ZBLS.

I like your steps. I would probably do two dedges in D during EOline to transition easier.
As in pairing two random dedges on D while pairing DF and DB ?

#### Lazy Einstein

##### Alg Ninja
As in pairing two random dedges on D while pairing DF and DB ?
Yeah. The same way you do cross dedges for Hoya.
Only with this variation of Hoya, you can do any non U layer edge instead of only the cross dedges in D.

#### Berkmann18

##### Member
Yeah. The same way you do cross dedges for Hoya.
Only with this variation of Hoya, you can do any non U layer edge instead of only the cross dedges in D.
I like this idea, it's more efficient than what I do at that stage.

#### AlphaSheep

##### Member
I pair 3 other edges together with the line edges. I feel that there are several advantages to doing steps 2-5 a y rotation away from my normal ZZ orientation, so I put my line edges on DR and DL and put the 3 other edges oriented at DB, BR and BL.

Edit: I found my post describing how I do Hoya for ZZ which includes some example solves:

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#### Lazy Einstein

##### Alg Ninja
I pair 3 other edges together with the line edges. I feel that there are several advantages to doing steps 2-5 a y rotation away from my normal ZZ orientation, so I put my line edges on DR and DL and put the 3 other edges oriented at DB, BR and BL.

Edit: I found my post describing how I do Hoya for ZZ which includes some example solves:
"Orientation parity can be fixed here using a slightly shorter alg than the standard one."
This is from your post. What alg do you use?
Also I am going to look into your variation and Max's and see what I can come up with.

#### AlphaSheep

##### Member
"Orientation parity can be fixed here using a slightly shorter alg than the standard one."
This is from your post. What alg do you use?
Also I am going to look into your variation and Max's and see what I can come up with.
The alg I'm using at the moment is this one:
x r U2 r' U2 l U2 r' U2 r D2 r D2 r' D2

#### Lazy Einstein

##### Alg Ninja
The alg I'm using at the moment is this one:
x r U2 r' U2 l U2 r' U2 r D2 r D2 r' D2