Date: Sun, 02 Jun 96 00:17:03 -0400 (EDT)
From: Dan Hoey <hoey@aic.nrl.navy.mil>
To: Cube Lovers
Subject: Re: Compact Cube Representation for Shamir and Otherwise

I'm not sure this is so interesting to all of cube-lovers; e-mail me
if you have opinions pro or con.

Jerry writes of the standard S24 x S24 model, which uses 48 bytes per
position without packing.  He also has a "supplement" representation
that uses one facelet from each edge and corner, for 20 bytes.  He
packs them into 13 bytes on tape.

The way I did it the last time I worked on brute force was to
pack eight twelve-bit fields:

  The orientations in two twelve-bit fields (2^11 and 3^7),
  The edge permutation in four twelve-bit fields,
  each of three base-12 digits (12^3), and
  The corner permutation in two twelve-bit fields, each of
  four base-8 digits (8^4).

Unpacking the fields can be done with native arithmetic or table
lookup.  In the latter case, it is better to use 12*11*10 instead of
12^3 and 8*7*6*5 instead of 8^3.

Also, postmultiplying by a fixed permutation can be done with table
lookup without unpacking.  I used this feature for twelve permutations
of particular interest.

I am somewhat rusty on the implications of using this representation
in conjunction with Shamir's algorithm.  I think it provides an
ordering of the permutations that enables at least an approximation
to the random access you need, then you unpack it and do a better job.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil