Archives

In FTM the complete knowledge of the distribution of the symmetric subgroups (first table of http://kociemba.org/symmetric2.htm ) lets us not only give the odd and even entries in the distance table but also the distances mod 48, because all unsymmetric position contribute with a multiple of 48. So we get

distance   positions mod 48

   0              1
   1             18
   2              3
   3             24
   4             39
   5             12
   6             22
   7             12
   8             40
   9              3
  10              4
  11             20

I've been thinking about writing a program to calculate and count duplicate positions - roughly speaking, those positions that are half way through an identity.  What I have in mind will probably be a more time consuming program to write than I would prefer.  So I wonder if I could ask Herbert Kociemba and/or mdlazreg to post a little something about the programs they have already written to find identities.  It may well be that there is a much simpler approach to calculating duplicate positions than what I have in mind.

What I have in mind is an iterative deepening depth first search beginning at the Start position.  If that's all I did, the search would simply count 12ⁿ maneuvers for each distance from n, and it would not extract any useful information about how many duplicate positions there are for each n.  To solve these problems, I propose to store all the duplicate positions and not to store those positions that are not duplicate.  This would be for the quarter turn metric.  The program I have in mind would not be able to handle the face turn metric.