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**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.

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)

**Advantages:**

* 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.

**Distadantages:**

* 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

Last edited: Aug 28, 2010