# <U, R, F> statistics



## xyzzy (May 14, 2016)

I've been trying to write a 3×3×3 solver lately (as part of a 5×5×5 reduction solver) and here's some statistics I got along the way, which may be of general interest.

To solve only the corners, this is equivalent to solving just a 2×2×2 and the depth distribution for that is all over the internet, but is reproduced here for completeness sake:


Spoiler: corner depth distribution



total states: 3674160
distance 0: 1 node
distance 1: 9 nodes
distance 2: 54 nodes
distance 3: 321 nodes
distance 4: 1847 nodes
distance 5: 9992 nodes
distance 6: 50136 nodes
distance 7: 227536 nodes
distance 8: 870072 nodes
distance 9: 1887748 nodes
distance 10: 623800 nodes
distance 11: 2644 nodes
average distance: 8.755576



And for the edges:


Spoiler: edge depth distribution



total states: 92897280
distance 0: 1 node
distance 1: 9 nodes
distance 2: 54 nodes
distance 3: 324 nodes
distance 4: 1944 nodes
distance 5: 11467 nodes
distance 6: 65400 nodes
distance 7: 356585 nodes
distance 8: 1800717 nodes
distance 9: 7801246 nodes
distance 10: 24692311 nodes
distance 11: 38040201 nodes
distance 12: 18320849 nodes
distance 13: 1790002 nodes
distance 14: 16168 nodes
distance 15: 2 nodes
average distance: 10.724572



The two antipodes at 15f for the edge-only <U, R, F> group are given by U R U2 F2 R F' U R F2 U2 R F U R F' and its inverse.

My 3-gen optimal solver is way too slow for me to run it on loads of positions, so here's a sample of 100 positions. Note that optimal 3-gen solutions aren't necessarily optimal 6-gen solutions (e.g. the T-perm alg is 15 moves in <U, R, F>, but can be solved in only 11 moves 6-gen).


Spoiler: sampled full depth distribution



total samples: 100
distance 16: 5
distance 17: 11
distance 18: 48
distance >18: 36
average distance: >18.15



Or if we force the states to be in <U, R, F2>, but still allow F and F' in the solves:


Spoiler: sampled full depth distribution | EO solved



total samples: 100
distance 15: 2
distance 16: 4
distance 17: 22
distance 18: 56
distance >18: 16
average distance: >17.80



And for people into FMC, here are some more stats for either leaving a corner 3-cycle or all corners solved. (I don't think this has ever been computed before?) First, corner-only optimal solution:


Spoiler: corner depth distribution leaving corner 3-cycle



total states: 3674160
distance 0: 631 nodes
distance 1: 5529 nodes
distance 2: 31370 nodes
distance 3: 167726 nodes
distance 4: 743302 nodes
distance 5: 1812382 nodes
distance 6: 901717 nodes
distance 7: 11503 nodes
average distance: 4.925585



The whole cube (using a thousand samples, since solving to L3C was something like fifty times faster than fully solving):


Spoiler: sampled full depth distribution leaving corner 3-cycle



total samples: 1000
distance 11: 2
distance 12: 17
distance 13: 75
distance 14: 274
distance 15: 452
distance 16: 180
average distance: 14.697



Also, the difference between the lengths of the optimal solution and the edge-only optimal solution (using the same set of 1000 random states as above):


Spoiler



0: 2
1: 11
2: 77
3: 217
4: 379
5: 233
6: 68
7: 12
8: 1
average difference: 3.997


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