# ZZ F2L Move Count

#### Cride5

Does anyone know the average and maximum move count for completing ZZ F2L (from EOLine) optimally, using <R,U,L> moves?

I found some statistics for completing F2L from EOCross here, but would be interested to know how many moves it takes from EOLine.

If possible I'd like the averages for F2L broken down in the following ways:

(1) L & R 1x2x3 blocks completed simultaneously
(2) Entire 1x2x3 L block, followed by entire 1x2x3 R block
(3) BL 1x2x2 followed by FL 1x1x2, then BR 1x2x2 followed by FR 1x1x2
(4) BL 1x2x2 followed by BR 1x2x2, then FL 1x1x2 followed by FR 1x1x2

Cheers.

EDIT: Also, how does it compare with the optimal move count using <L,R,U,D,F2,B2>?

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#### rjohnson_8ball

##### Member
You cannot flip an edge using only <R,U,L> moves. Wouldn't you need to do at least one cube rotation or an r,u,l,F,B,M,E, or S? Sorry I cannot help with your primary questions.

Edit: Oops, sorry, I didn't know what EOLine meant before I posted.

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#### Cride5

You cannot flip an edge using only <R,U,L> moves. Wouldn't you need to do at least one cube rotation or an r,u,l,F,B,M,E, or S? Sorry I cannot help with your primary questions.
I'm looking for the statistics to complete F2L from EOLine (edges already oriented and DF and DB placed). EOLine takes ~6.127 moves avg and 9 moves max (FTM) to do optimally.

#### Cride5

...looks like another coding job, but I'll have to put it on the back burner for now, too many other things to do! If anyone can find these stats in the mean time I'd be very grateful, along with a few other ZZ'ers

EDIT: Just for some quick info on the task involved, these are the numbers of cases which need to be solved:
(1) L + R blocks: 3^4 * (8!/4!) * (10!/4!) = 20,575,296,000 <-- a biggie
(2) L: 8*7 * 10*9*8 * 3^2 = 362,880 R: 6*5 * 7*6*5 * 3^2 = 56,700 (419,580 total)
(3) BL 1x2x2: 8 * 10*9 * 3 = 2160, FL 1x1x2: 7 * 8 * 3 = 168, BR 1x2x2: 6 * 7*6 * 3 = 756, FR 1x1x2: 5 * 5 * 3 = 75 (3159 total)
(4) BL 1x2x2: 8 * 10*9 * 3 = 2160, BR 1x2x2: 7 * 8*7 * 3 = 1176, FL 1x1x2: 6 * 6 * 3 = 108, FR 1x1x2: 5 * 5 * 3 = 75 (3519 total)

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#### Lt-UnReaL

##### Member
(3) BL 1x2x2: 8 * 10*9 * 3 = 2160, FL 1x1x2: 7 * 8 * 3 = 168, BR 1x2x2: 6 * 7*6 * 3 = 756, FR 1x1x2: 5 * 5 * 3 = 75 (3159 total)
Coincidence?

#### Cride5

(3) BL 1x2x2: 8 * 10*9 * 3 = 2160, FL 1x1x2: 7 * 8 * 3 = 168, BR 1x2x2: 6 * 7*6 * 3 = 756, FR 1x1x2: 5 * 5 * 3 = 75 (3159 total)
Coincidence?
Yup. There is a reason the total number of cases for (3) and (4) are different. The number of cases depends on (a) the number of pieces involved and (b) the number of possible orientations/permutations of these pieces.

Ignoring the first and last cases which have equal case counts: In (4) doing the 1x2x2 first means that (a) and (b) are both maximal and then both minimal when multiplied together, where as in (3) when (a) is maximal (b) is not and vice versa, yielding a lower overall total.

#### Cride5

That's great, cheers ... edging closer to the goal

So based on 100 random scrambles the statistics for strategy (4) is ~24.64 moves or ~30.77 including EOLine.

You wouldn't happen to have kept the averages for each sub-step would you?

#### Stefan

##### Member
So based on 100 random scrambles the statistics for strategy (4) is ~24.64 moves or ~30.77 including EOLine.
No. Your (4) solves the four 2x2x1 blocks in a specific order. I didn't. Always solved the shortest one first.

You wouldn't happen to have kept the averages for each sub-step would you?
Looks like I didn't keep them. Can compute it again though, but after the world champs.

#### miniGOINGS

##### Member
I might do this with 100 different scrambles and post my average movecount for (1), (2), (3), and (4).

#### Cride5

I might do this with 100 different scrambles and post my average movecount for (1), (2), (3), and (4).
Hi that would be a great help, thanks. Could you also use this or something like it for your scrambles? Most online scramblers don't actually create statistically random cube states. See this thread for details..

Cheers

#### miniGOINGS

##### Member
I might do this with 100 different scrambles and post my average movecount for (1), (2), (3), and (4).
Hi that would be a great help, thanks. Could you also use this or something like it for your scrambles? Most online scramblers don't actually create statistically random cube states. See this thread for details..

Cheers
Although I have a few questions:

1. Would this be without EOline? (I probably would switch to ZZ, at least for OH, if I could figure out a fast way of doing EOline) Yes.

2. Should I use fixed colours to reduce the chance of luck?

3. Should I include the solutions?

EDIT: I answered my first question by reading your post. My move count for (1) is probably going to be really bad just because I don't have any practice with it.

EDIT2: So I just realized that when doing ZZ F2L, I pretty much always do either (3) or (4) and even when I try to do (1) or (2) it turns out to be (3) or (4).

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#### Cride5

I might do this with 100 different scrambles and post my average movecount for (1), (2), (3), and (4).
Hi that would be a great help, thanks. Could you also use this or something like it for your scrambles? Most online scramblers don't actually create statistically random cube states. See this thread for details..

Cheers
Although I have a few questions:

1. Would this be without EOline? (I probably would switch to ZZ, at least for OH, if I could figure out a fast way of doing EOline) Yes.

2. Should I use fixed colours to reduce the chance of luck?

3. Should I include the solutions?

EDIT: I answered my first question by reading your post. My move count for (1) is probably going to be really bad just because I don't have any practice with it.

EDIT2: So I just realized that when doing ZZ F2L, I pretty much always do either (3) or (4) and even when I try to do (1) or (2) it turns out to be (3) or (4).
Sorry I misunderstood you, I didn't realise you were planning on doing this manually! We're looking for optimal move counts. Doing this manually for (3) and (4) is possible (if you really know what you're doing), but for (1) and (2) you really need to be using a computer solver which guarantees optimality. The only ones I know of are:
Graphical: Cube Explorer in huge solver mode (windoze only)
Command line: ACube, this, this and this.

Generally I wouldn't recommend tackling this kind of problem manually.. even if you find a good solution its very difficult to prove with absolute certainty that its an optimal one. Because its possible to do an exhaustive search with a computer solver, we can guarantee that solutions are optimal..

The added benefit of a computer is that its a **** load faster then even the best human solvers we know

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#### miniGOINGS

##### Member
Sorry I misunderstood you, I didn't realise you were planning on doing this manually! We're looking for optimal move counts. Doing this manually for (3) and (4) is possible (if you really know what you're doing), but for (1) and (2) you really need to be using a computer solver which guarantees optimality. The only ones I know of are:
Graphical: Cube Explorer in huge solver mode (windoze only)
Command line: ACube, this, this and this.

Generally I wouldn't recommend tackling this kind of problem manually.. even if you find a good solution its very difficult to prove with absolute certainty that its an optimal one. Because its possible to do an exhaustive search with a computer solver, we can guarantee that solutions are optimal..

The added benefit of a computer is that its a **** load faster then even the best human solvers we know
Yea, I planned on doing it manually. Nevermind then I guess, it would take me too long to use program, sorry.

#### Cride5

In order to clear up some recent discussions on ZZF2L move count, I've generated some quick stats for strategy (2) - solving an entire 1x2x3 block at a time. It was done using Johannes's web solver.

I took 100 samples based on solving the easiest 1x2x3 block first. The average optimal move count for both blocks (excluding EOLine) was: 18.72 HTM
Adding the average of ~ 6.13 for EOLine gives a total of 24.85 for all of F2L.

Actual data:
16, 16, 17, 19, 18, 21, 17, 19, 21, 20, 19, 18, 21, 17, 19, 16, 19, 14, 20, 19, 20, 18, 17, 18, 21, 19, 19, 19, 18, 19, 20, 15, 21, 17, 19, 20, 19, 18, 18, 13, 17, 19, 17, 18, 19, 19, 19, 18, 18, 19, 20, 20, 19, 18, 19, 20, 20, 20, 17, 20, 20, 18, 19, 19, 17, 19, 20, 21, 22, 19, 18, 20, 19, 21, 17, 20, 17, 19, 19, 19, 18, 20, 17, 20, 20, 18, 21, 19, 19, 21, 18, 16, 20, 20, 18, 18, 18, 20, 20, 18
This provides an estimate for the absolute lower bound for ZZ F2L. Data from Stephan shows that ZZ F2L takes around ~30.77 optimally if each side is broken down into 1x1x2 and 1x2x2 blocks and solved in easiest-first order.

In reality a human solver is rarely going to be able to solve an entire side optimally, but with experience a solver may sometimes solve an entire side or employ other tricks to achieve a lower move count than by breaking F2L down into four blocks.

So to compare with Fridrich:
Cross + 4xSlots: 5.81 + 23.03 = 28.84
CN Cross + 4xSlots: 4.81 + 23.03 = 27.84
EOCross + 4xSLots <R,U,L>: 7.53 + 28.1 = 35.63
EOLine + 4x F2L Blocks: 6.13 + 24.64 = 30.77
EOLine + 2x F2L Blocks: 6.13 + 18.72 = 24.85

EDIT: Thanks to cuBerBruce stats I've added EOCross..

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#### a small kitten

##### Member
EOLine + 4x F2L Blocks: 6.13 + 24.64 = 30.77
EOLine + 2x F2L Blocks: 6.13 + 18.72 = 24.85
Does that mean you are saying that the EO line takes as many moves as the EO cross?

#### Cride5

No, not at all! I didn't post any figures for EOCross.

Just to clarify what I meant:

* Cross + 4xSlots = Just regular Fridrich - cross followed by four slots.

* CN Cross + 4xSlots = Just regular Fridrich but colour neutral.

* EOLine + 4x F2L Blocks = EOLine, then in easiest first order: RH 1x2x2, LH 1x2x2, RH 1x1x2, LH 1x1x2 (ensuring the 1x2x2 comes before the 1x1x2 on each side)

* EOLine + 2x F2L Blocks = EOLine, then in easiest first order: entire RH 1x2x3, entire LH 1x2x3

The optimal move count for F2L based on EOCross is pretty bad. From this thread, excluding EO its an average of 32.2 moves. I don't know the figures for optimal EOCross, but assuming it is something like 7-8 moves that would make EOCross-based F2L over 35 moves avg.

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#### cuBerBruce

##### Member
For what it's worth, I've computed optimal distributions for adding left side 1x2x3 from EOLine in both <R,U,L> and <U,D,R,L,F2,B2> (face-turn metric). I note that this if for a specific 1x2x3 block, and not for the easier of the left and right block. The average distance for <R,U,L> is about 9.6922, while the average distance for <U,D,R,L,F2,B2> is about 8.4048.

Code:
distance  <R,U,L>   <U,D,R,L>  <U,D,R,L,F2,B2>
0           1           1           1
1           3           3           3
2           9           9           9
3          48          53          67
4         182         236         363
5         769        1122        1842
6        2938        4862        9169
7       10783       19743       39784
8       35330       66495      122543
9       89070      145949      165797
10      136195      113465       23277
11       77681       10907          25
12        9769          35
13         102
I adapted my optimal 2x2x3 block program to generate these results. For <U,D,R,L,F2,B2>, I did a breadth-first search to generate all 47900160 reachable positions of the 7 cubies, and then filtered the result to only those corresponding to EOLine positions.

EDIT: I've added <U,D,R,L> data (requested by Stefan) to the above table. The average move count for <U,D,R,L> came out to about 9.0241.

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