# First center on 5x5x5



## cuBerBruce (Mar 7, 2013)

There has been breadth first searches done for various 3x3x3 first steps, and first two centers (opposite faces) on 4x4x4. I don't recall seeing any similar analysis for the 5x5x5. So I've done breadth first searches for solving the first center (4 +-centers and 4 x-centers put on the same face of the matching central face piece).

For outer block turns only, I found it takes no more than 10 moves.

For single-layer turns only, I found it takes no more than 9 turns.

Combining these gets the maximum move count down to 8 moves.

Allowing all 90 block turns, 8 moves are still required.

Naturaly, for my analyses, I implemented inner turns involving a central layer as if they are two outer block turns with the same effect other than the orientation of the cube, so the the central face pieces remain fixed reference pieces. This basically just makes the calculation simpler.


Spoiler





```
Outer block turns only:
distance to solved state  0: count =         1, total         1
distance to solved state  1: count =        12, total        13
distance to solved state  2: count =       190, total       203
distance to solved state  3: count =      3210, total      3413
distance to solved state  4: count =     47444, total     50857
distance to solved state  5: count =    635642, total    686499
distance to solved state  6: count =   6870327, total   7556826
distance to solved state  7: count =  41671278, total  49228104
distance to solved state  8: count =  59387592, total 108615696
distance to solved state  9: count =   4296076, total 112911772
distance to solved state 10: count =       104, total 112911876

Single-layer turns only:
distance to solved state  0: count =         1, total         1
distance to solved state  1: count =        18, total        19
distance to solved state  2: count =       343, total       362
distance to solved state  3: count =      6842, total      7204
distance to solved state  4: count =    120148, total    127352
distance to solved state  5: count =   1799154, total   1926506
distance to solved state  6: count =  18389656, total  20316162
distance to solved state  7: count =  67300794, total  87616956
distance to solved state  8: count =  25242532, total 112859488
distance to solved state  9: count =     52388, total 112911876

Wide turns (face layer with next layer) and all single-layer turns
distance to solved state  0: count =         1, total         1
distance to solved state  1: count =        18, total        19
distance to solved state  2: count =       407, total       426
distance to solved state  3: count =     10038, total     10464
distance to solved state  4: count =    218116, total    228580
distance to solved state  5: count =   4023244, total   4251824
distance to solved state  6: count =  41125814, total  45377638
distance to solved state  7: count =  65857324, total 111234962
distance to solved state  8: count =   1676914, total 112911876

All block turns (90 allowed moves)
distance to solved state  0: count =         1, total         1
distance to solved state  1: count =        18, total        19
distance to solved state  2: count =       471, total       490
distance to solved state  3: count =     13474, total     13964
distance to solved state  4: count =    343032, total    356996
distance to solved state  5: count =   7181482, total   7538478
distance to solved state  6: count =  62806446, total  70344924
distance to solved state  7: count =  42524996, total 112869920
distance to solved state  8: count =     41956, total 112911876
```


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## Noahaha (Mar 7, 2013)

I might not understand how you did it, but thanks for putting a number on it! Now I know what to work towards.


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## Akash Rupela (Mar 7, 2013)

is this number for any particular face or is this color neutral?


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## stannic (Mar 7, 2013)

Wow, congratulations for the first 5x5x5-related analysis.

Do you have any plans on performing multi-stage analysis of the 5x5x5 similar to one you did for 4x4x4?
If so, could you suggest any sequence(s) of nested substages which is (are) better suitable for such analysis (as you did for 4x4x4)?


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## cuBerBruce (Mar 7, 2013)

Akash Rupela said:


> is this number for any particular face or is this color neutral?


It's for solving one selected face's set of center pieces. An exact distance distribution for color neutral is pretty much beyond reach with today's technology, but a good approximation would be doable.


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## ben1996123 (Mar 7, 2013)

cuBerBruce said:


> It's for solving one selected face's set of center pieces. An exact distance distribution for color neutral is pretty much beyond reach with today's technology, but a good approximation would be doable.



Does single colour/colour neutral actually make a difference on centers, since there is only 1 sticker on each piece? every sticker of the centers is independent of all the others, unlike 3x3 cross.


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## cuBerBruce (Mar 7, 2013)

stannic said:


> Wow, congratulations for the first 5x5x5-related analysis.
> 
> Do you have any plans on performing multi-stage analysis of the 5x5x5 similar to one you did for 4x4x4?
> If so, could you suggest any sequence(s) of nested substages which is (are) better suitable for such analysis (as you did for 4x4x4)?



Well, thanks. But this calculation only involved around a hundred million positions. A breadth-first search of this size (without symmetry reduction) only takes a few minutes of cpu time to run. This analysis was smaller than any of the five stages of my 4x4x4 analysis.

I have not really planned to do a multi-phase analysis of 5x5x5 at this time. Perhaps I can look at possibilities for choices of nested subgroups.



ben1996123 said:


> Does single colour/colour neutral actually make a difference on centers, since there is only 1 sticker on each piece? every sticker of the centers is independent of all the others, unlike 3x3 cross.


Each center color would have the same distribution, but by color neutral, I would understand that we would need to consider scrambles of all 48 moveable center pieces, and then (using a table such as produced by my breadth-first searches) look up how many moves are required for each of the six faces, and picking the smallest. Doing this for thousands or millions of random positions would allow getting an approximate value for color neutral.


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