# 2x2x2, X-cross and 3x2x2 move count?



## JL58 (May 29, 2009)

I am getting bored with regular Fridrich on 3^3. I am starting to play around with block building and X-cross. I found it very interesting. My move count, however, is erratic at best. I was wondering if anyone has computed the average and upper bound move counts for building:

- an X-cross
- a 2x2x2 block
- a 3x2x2 block

I am looking for this under 2 conditions: color agnostic solve and using a pre-defined color for the cross or the blocks. So I am asking for 12 numbers in total.


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## cuBerBruce (May 29, 2009)

See http://cubezzz.homelinux.org/drupal/?q=node/view/116 for 2x2x2 and 3x2x2. (Actually I was not the first to do it for 2x2x2.) I don't know if values are known for when choosing among all possible 2x2x2 blocks, but it is known that worst case for choosing among all possible 3x2x2 blocks is 11 moves FTM.


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## Johannes91 (May 29, 2009)

cuBerBruce said:


> I don't know if values are known for when choosing among all possible 2x2x2 blocks


I solved 3 million scrambles a couple of years ago (link), and the average was 4.870 moves. If I remember correcrly, I used 25 move scrambles, and the move count was very slightly higher for longer or perfect scrambles.

Worst case is 8 moves (link).

For color-fixed XCross, average is 7.976 and worst case 10 moves (link). Note that this is using the same cross *and* same ce-pair every time.

I'll do some averages for color agnostic XCross and 2x2x3 and post here later.


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## Robert-Y (May 29, 2009)

Johannes91 said:


> cuBerBruce said:
> 
> 
> > I don't know if values are known for when choosing among all possible 2x2x2 blocks
> ...



What's a perfect scramble?


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## rahulkadukar (May 29, 2009)

Exactly what is a perfect scramble and what is 4.87 HTM or QTM


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## Neutrals01 (May 29, 2009)

HTM - Half turn metric(Half turn<180 degrees> counted as 1 move)
QTM - Quarter turn metric(Quarter turn<90 degrees> counted as 1 move)


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## Johannes91 (May 29, 2009)

Robert-Y said:


> What's a perfect scramble?


One that gives every position an equal probability. Doing a finite number of random moves doesn't, generating a random position does.

No need to quote the whole post when you're only replying to a small part.


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## Robert-Y (May 29, 2009)

Johannes91 said:


> Robert-Y said:
> 
> 
> > What's a perfect scramble?
> ...



Hmmm.... How many perfect scrambles are there? Also, can you give me an example of a perfect scramble, just to make sure I understand what you mean please?


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## Johannes91 (May 29, 2009)

I should've said "perfect scrambling method" instead of "perfect scramble".

Scrambling method #1: Start with a solved cube and make N random moves.
Some positions are more probable than others. The differences get smaller as N gets larger.

Scrambling method #2: Generate a random position.
Assuming the random number generator is good, this gives every position an equal probability.

Scrambling has been discussed many times here and in yahoo groups, search if you're interested.

Yay, 6 off-topic posts in a row.


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## Johannes91 (May 30, 2009)

```
| Average | Worst  | Scrambles
----------------------+---------+--------+-----------
XCross color neutral  | ~6.536  | [9,10] | 2^26
XCross opposite cross | ~6.986  |   10   | 2^28
XCross fixed cross    | ~7.354  |   10   | 2^29
XCross fixed          |  7.976  |   10   | n/a
----------------------+---------+--------+-----------
2x2x3 color neutral   | ~7.991  |   11   | 2^21
2x2x3 fixed           |  9.116  |   12   | n/a
----------------------+---------+--------+-----------
2x2x2 color neutral   | ~4.916  |    8   | 2^28
2x2x2 fixed           |  6.034  |    8   | n/a
----------------------+---------+--------+-----------
1x2x3 color neutral   | ~5.144  |  [8,9] | 2^26
1x2x3 LD|RD|LU|RU     | ~5.943  |    9   | 2^27
1x2x3 LD|RD           | ~6.306  |    9   | 2^28
1x2x3 fixed           |  6.701  |    9   | n/a
----------------------+---------+--------+-----------
1x2x2 color neutral   | ~2.671  |    5   | 2^25
1x2x2 fixed           |  4.360  |    5   | n/a
----------------------+---------+--------+-----------
EOLine color neutral  | ~5.006  |    9   | 2^25
EOLine fixed          |  6.127  |    9   | n/a
```

- All move counts are in HTM/FTM.
- I used random state scrambling.
- Color neutral numbers are only approximations.

Thanks to Bruce for the fixed color 2x2x3 numbers.

Edit 2009-06-04: Added 1x2x3 block a.k.a. Roux block.
Edit 2009-07-06: FYI, the distributions are at the end of this post.
Edit 2009-08-01: Added two partially neutral 1x2x3 blocks and 1x2x2 block.
Edit 2009-08-09: Added EOLine.


```
(total positions, mean, list of (depth, positions at depth) pairs)

XCross
(67108864,6.535903126001358,[(0,20),(1,315),(2,3691),(3,40391),(4,429582),(5,3973663),(6,24007367),(7,36373364),(8,2280468),(9,3)])
(268435456,6.986002378165722,[(0,27),(1,440),(2,4895),(3,56666),(4,620074),(5,6130185),(6,47023196),(7,157020549),(8,57518550),(9,60874)])
(536870912,7.353692278265953,[(0,20),(1,441),(2,5044),(3,56557),(4,624256),(5,6206267),(6,50307192),(7,232710047),(8,239188357),(9,7772726),(10,5)])
(72990720,7.975721804086875,[(0,1),(1,15),(2,172),(3,1950),(4,21535),(5,220368),(6,1989591),(7,13431990),(8,40963892),(9,16325184),(10,36022)])
2x2x3
(2097152,7.990720748901367,[(1,2),(3,36),(4,358),(5,3838),(6,40971),(7,350772),(8,1274997),(9,425962),(10,216)])
(1532805120,9.115899440628173,[(0,1),(1,12),(2,141),(3,1746),(4,20935),(5,243092),(6,2698935),(7,27258179),(8,216204042),(9,830686751),(10,453825501),(11,1865784),(12,1)])
2x2x2
(268435456,4.915635708719492,[(0,8271),(1,74990),(2,757290),(3,7061775),(4,52689991),(5,161090773),(6,46724373),(7,27993)])
(253440,6.033834438131313,[(0,1),(1,9),(2,90),(3,852),(4,7169),(5,44182),(6,131636),(7,68940),(8,561)])
1x2x3
(67108864,5.144310295581818,[(0,1192),(1,10717),(2,111293),(3,1126409),(4,8975643),(5,35710034),(6,21051484),(7,122092)])
(134217728,5.9427317678928375,[(0,400),(1,3575),(2,37944),(3,399141),(4,3678904),(5,25824316),(6,77456888),(7,26767934),(8,48626)])
(268435456,6.306151166558266,[(0,365),(1,3679),(2,37788),(3,399662),(4,3733678),(5,28185912),(6,120408288),(7,112126360),(8,3539713),(9,11)])
(5322240,6.700925174362674,[(0,4),(1,36),(2,376),(3,3976),(4,37140),(5,289044),(6,1495720),(7,2885332),(8,609716),(9,896)])
1x2x2
(33554432,2.6706851422786713,[(0,248598),(1,1443471),(2,9779999),(3,19719644),(4,2362678),(5,42)])
(12672,4.360164141414141,[(0,4),(1,24),(2,208),(3,1388),(4,5104),(5,5432),(6,512)])
EOLine
(33554432,5.006114274263382,[(0,1496),(1,12273),(2,112415),(3,967997),(4,6226405),(5,17516431),(6,8673453),(7,43962)])
(270336,6.126535126657197,[(0,1),(1,9),(2,91),(3,851),(4,6831),(5,41703),(6,130239),(7,88683),(8,1927),(9,1)])
```


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## Weston (May 30, 2009)

you shouldnt completely devote yourself to one single method and just keep your options open and just use X-cross when its convenient

expecially if your not color netural.

it would help though it you could to opposite color crosses, when you could build the 2x2x2 block anywhere and just either finish the cross one of the 2 sides, rather then all 6 sides


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## cmhardw (May 30, 2009)

Johannes, what are the numbers for solving X-cross on either of two opposite colors, with preference of color being always toward the easiest of the two, and not fixing the slot? I have a theory that this is when using Xcross is the most practical, and this is how I use X-cross. I'd be curious to see the data behind this particular approach.

Chris

--edit--
Using this approach I feel I should mention that I start by solving an X-cross between 40%-50% of my solves. I feel such a strategy allows you to use Xcross frequently (decently high probability to have a somewhat easy case) without having to learn cases or study patterns via memorization. I still do my Xcross solving intuitively and by block building.
--edit--


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## Johannes91 (May 30, 2009)

cmhardw said:


> Johannes, what are the numbers for solving X-cross on either of two opposite colors, with preference of color being always toward the easiest of the two, and not fixing the slot?


Worst case is 10 moves, for example superflip with swaps (UF UB) and (DF DB). Average seems to be a bit below 7.


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## Johannes91 (Jun 14, 2009)

Some more data than just average and worst case:
OMG I'm sorry for bumping a 2-week-old topic! I don't even have a question!
XCross. If you could easily find 6-movers but rarely 7-movers, with full color neutrality you could use XCross in ~42% of your solves; with opposite cross, ~20%; and with fixed cross, only ~11%.






2x2x2 block.





2x2x3 block.





First Roux block a.k.a. 1x2x3 block. The difference is quite big because there are 24 options a color neutral cuber can choose from. AFAIK, many cubers are partially neutral when using Roux.


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## pjk (Jun 14, 2009)

Johannes91 said:


> Some more data than just average and worst case:
> OMG I'm sorry for bumping a 2-week-old topic! I don't even have a question!
> XCross. If you could easily find 6-movers but rarely 7-movers, with full color neutrality you could use XCross in ~42% of your solves; with opposite cross, ~20%; and with fixed cross, only ~11%.
> [images]


I am not quite understanding the data. Shouldn't color neutrality always give you the highest percentage for each number of moves? For example, for the XCross plot, at 7 moves, opposite color has a higher percentage than color neutral. This doesn't quite make sense because color neutral includes opposite color, so the percentage of color neutral should be _at least _the percentage of opposite color.

*Edit:* Ah, it is based on percentage of all moves, and not just each move individually. Obvious now.


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## Kian (Jun 14, 2009)

pjk said:


> Johannes91 said:
> 
> 
> > Some more data than just average and worst case:
> ...



However, if you look at 6 and 5 moves, there are many, many more possibilities for fully color neutral. The percentage of *remaining* cases for color neutral at 7 moves is higher, but not the gross, because there are simply very few 5 and 6 solves for opposite cross.

If that graph is to scale, then it seems that over 90% of color neutral x-crosses can be done in 7 moves or less, but a considerably smaller percentage can be done with opposite cross in 7 moves or less.

Of course, the graph could be inaccurate, but I think that makes sense.


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## JL58 (Jun 18, 2009)

Johannes91, Kian and all, I want to thank you for your responses. Very informative. I loved the opposite cross data - a favorite (it seems) of Erik the half God.

I am still baffled by the results. I wonder how many of the CFOP disciples are close enough to find the optimum Xcross within normal inspection time. 

This data is encouraging. Working on Xcross would save 25% of the pair insertion time with almost no move cost and is therefore worth a lot.

In general I find CFOP (like probably all other methods) very compelling as progress relies on shifting the automatic moves to the more creative steps (algorithmic pair insertion to intuitive, 4 pair insertions to Xcross, etc.) - obvious to many of you, a learning for the rest of us.

Thanks again for your support.


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## JLarsen (Jun 18, 2009)

How close to these moves can people actually get during a speedsolve currently?


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