God's Algorithm, Face Turn Metric, Out to 11 Moves from Start

Distance   Patterns Unique     Positions
 from      up to Symmetry

  0                     1               1
  1                     2              18
  2                     9             243
  3                    75            3240      
  4                   934           43239      
  5                 12077          574908      
  6                159131         7618438      
  7               2101575       100803036      
  8              27762103      1332343288      
  9             366611212     17596479795      
 10            4838564147    232248063316      
 11           63818720716   3063288809012 
Jerry Bryan

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Congratulations! The numbers

Congratulations! The numbers look incredibly huge although I probably should not be surprised. How long did this take?
Did you use only one computer?


One computer. The run took a

One computer. The run took a couple of months. In round numbers, the program calculates about 15,000 patterns per second, roughly equivalent to about 720,000 positions per second. It doesn't actually calculate positions. It only calculates patterns. For each pattern, it determines the number of equivalent positions based on the symmetry characteristics of the pattern. I forget the exact model of the computer, but it's a standard Windows desktop, about 3 1/2 years old, so I think the Pentium chip runs at about 2.6Ghz or so.

I use the word "pattern" to denote a respresentative element of a position under symmetery, where for a position x, Rep(x)=min(xm), for all m in M, and where M is the set of 48 rotations and reflections of the cube. So for y=Rep(x), x is the position and y is the pattern. Of course, if y is a pattern, then it is the case that y=Rep(y).

Unfortunately, my program currently does not calculate antisymmetry.