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Date: Fri, 15 Aug 1997 19:00:26 -0400
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From: Dan Hoey <Hoey@aic.nrl.navy.mil>
To: reid@math.brown.edu
Cc: cube-lovers@ai.mit.edu
In-Reply-To: <199708142216.SAA16395@life.ai.mit.edu> (message from michael
	reid on Thu, 14 Aug 1997 18:21:24 -0400)
Subject: Re: patterns with 24-fold symmetry

Mike Reid writes:

> i've finished computing minimal maneuvers for those positions with
> 24-fold symmetry.  these positions were classified by dan hoey and
> jim saxe in their note "symmetry and local maxima."  there are 24
> such positions; they form an abelian subgroup of type 6, 2, 2.

It took me a while to understand that.  For the benefit of other
cube-lovers, since any finite Abelian group can be decomposed into a
direct product of cyclic groups, it can be typified by listing the
orders of its factors.

> we may take as generators  superfliptwist, pons asinorum, and 6 H's.
> of these 24 positions, 4 have 48-fold symmetry; i'll include these
> here as well.  the other 20 positions occur in 10 pairs which differ
> only in orientation; i.e. there are 10 "patterns".

It may be better to take the order-6 generator to be one of the
6-H-supertwists.  Then you can tell the M-symmetric positions because
they project to the identity of the 6-factor.  Writing p, f, t, h for
pons, superflip, supertwist, and 6-H, I get the following table of
positions (suffixed with optimal qtw:ftw).

        i          p          f            fp
      ...............................................
    i : i    0:0   p   12:6   f    24:20   fp   20:19
      :
   th : th  20:16  th' 20:16  fth  22:20   fth' 22:20
      :
    t : t   22:16  pt  20:16  ft   20:20   fpt  22:20
      :
    h : h   16:8   h   16:8   fh   18:17   fh   18:17
      :
    t : t   22:16  pt  20:16  ft   20:20   fpt  22:20
      :
   th': th' 20:16  th  20:16  fth' 22:20   fth  22:20

The last two rows could be omitted, just as the last column could be
with your decomposition:

        i          h            p          h
      ...............................................
    i : i   0:0    h    16:8    p   12:6   h    16:8
      :
   ft : ft 20:20   fth  22:20   ftp 22:20  fth' 22:20
      :
    t : t  22:16   th   20:16   tp  20:16  th'  20:16
      :
    f : f  24:20   fh   18:17   fp  20:19  fh   18:17
      :
    t : t  22:16   th'  20:16   tp  20:16  th   20:16
      :
   ft : ft 20:20   fth' 22:20   ftp 22:20  fth  22:20

This has the advantage of having patterns on each row nearer each
other.

By the way, this isn't a complete list of optimal maneuvers, is it?
Are you looking to find such a list?  Or would it be too difficult (or
too voluminous)?

And I'm looking forward to seeing optimal maneuvers for the
T-symmetric positions (if I'm not being too presumptuous).

Dan
Hoey@AIC.NRL.Navy.Mil