Date: 30 Jul 1987 15:46:10 EDT (Thu)
From: Dan Hoey <hoey@nrl-aic.ARPA>
Subject: Planar positions of Rubik's Magic
To: Cube-Lovers@AI.AI.MIT.EDU
Message-Id: <554672771/hoey@nrl-aic>

    PLANAR POSITIONS OF RUBIK'S MAGIC, THE 8 SQUARE PUZZLE

    by P Beck and D Hoey,  July 1987
       <BECK@ARDEC-LCSS.ARPA> or <PBECK@ARDEC.ARPA>,
       <HOEY@NRL-AIC.ARPA>

This is a catalog of the 96 planar positions of the 8-square Rubik's
Magic puzzle.  The list is based on two rules for positioning the eight
squares.

RULE 1--Placement:  Let the pieces be numbered from 1 to 8.  Any planar
position must consist of squares in the pattern

                ``2x4''       or      ``3x3''

                A B C D                A B C
                H G F E                H E D
                                       G F

where A,B,C,D,E,F,G,H is a cyclical rearrangement of 1,2,3,4,5,6,7,8.
These patterns can also be rotated or reflected.

Both the 2x4 and the 3x3 patterns have eight rotations and reflections,
and there are eight possible assignments of the numbers 1-8 to the
letters A-H.  However, a 180-degree rotation of the 2x4 is equivalent
to a reassignment of the numbers.  So there are only 32 different 2x4
positions, while there are a full 64 of the 3x3 positions.

RULE 2--Orientation:  The pieces fit together as if the four edges
of each unrotated piece were

         +-b-+                                +-d-+
labeled  a O c  for odd-numbered pieces, and  a E c  for even-numbered
         +-d-+                                +-b-+

pieces, and the small letters must match where neighbors abut.