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.

>From rule 1, it is apparent that when neighbors abut, one of them must
be an even-numbered piece and the other odd. From rule 2, we observe
that if a piece is rotated by 0 or 180 degrees, then its top and bottom
neighbors must be rotated the same amount and its left and right neigh-
bors must be rotated 180 degrees differently. In this catalog, piece 1
will be placed in its unrotated orientation. Then the orientation of
each piece is determined from its position relative to piece 1, and the
entire position is determined by the choice of pattern under rule 1.

TRANSFORMATION: Each 2x4 position can be directly transformed into any
of four 3x3 positions, by folding out either end to either side. Each
of the 3x3's can be directly transformed into either a vertical or a
horizontal 2x4.

SYMBOLOGY: Plain numbers indicate an unrotated piece, while numbers
followed by an asterisk indicate pieces rotated 180 degrees.

The use of numbers seems to be the most popular alternate graphics
pattern at this time, as it most clearly shows what is happening as the
puzzle is manipulated.

ACKNOWLEDGEMENT: Thanks to Rodney Hoffman for reviewing a preliminary
version of the catalog and the inspiration to prepare it in the first
place.

THE CATALOG:

>>>16 VERTICAL POSITIONS

+--+--+    +--+--+    +--+--+    +--+--+
|1 |8*|    |1 |2*|    |8*|1 |    |2*|1 |
+--+--+    +--+--+    +--+--+    +--+--+
|2 |7*|    |8 |3*|    |7*|2 |    |3*|8 |
+--+--+    +--+--+    +--+--+    +--+--+
|3 |6*|    |7 |4*|    |6*|3 |    |4*|7 |
+--+--+    +--+--+    +--+--+    +--+--+
|4 |5*|    |6 |5*|    |5*|4 |    |5*|6 |
+--+--+    +--+--+    +--+--+    +--+--+

+--+--+    +--+--+    +--+--+    +--+--+
|8 |7*|    |2 |3*|    |7*|8 |    |3*|2 |
+--+--+    +--+--+    +--+--+    +--+--+
|1 |6*|    |1 |4*|    |6*|1 |    |4*|1 |
+--+--+    +--+--+    +--+--+    +--+--+
|2 |5*|    |8 |5*|    |5*|2 |    |5*|8 |
+--+--+    +--+--+    +--+--+    +--+--+
|3 |4*|    |7 |6*|    |4*|3 |    |6*|7 |
+--+--+    +--+--+    +--+--+    +--+--+

+--+--+    +--+--+    +--+--+    +--+--+
|7 |6*|    |3 |4*|    |6*|7 |    |4*|3 |
+--+--+    +--+--+    +--+--+    +--+--+
|8 |5*|    |2 |5*|    |5*|8 |    |5*|2 |
+--+--+    +--+--+    +--+--+    +--+--+
|1 |4*|    |1 |6*|    |4*|1 |    |6*|1 |
+--+--+    +--+--+    +--+--+    +--+--+
|2 |3*|    |8 |7*|    |3*|2 |    |7*|8 |
+--+--+    +--+--+    +--+--+    +--+--+

+--+--+    +--+--+    +--+--+    +--+--+
|6 |5*|    |4 |5*|    |5*|6 |    |5*|4 |
+--+--+    +--+--+    +--+--+    +--+--+
|7 |4*|    |3 |6*|    |4*|7 |    |6*|3 |
+--+--+    +--+--+    +--+--+    +--+--+
|8 |3*|    |2 |7*|    |3*|8 |    |7*|2 |
+--+--+    +--+--+    +--+--+    +--+--+
|1 |2*|    |1 |8*|    |2*|1 |    |8*|1 |
+--+--+    +--+--+    +--+--+    +--+--+

>>16 HORIZONTAL POSITIONS

+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+
|1 |8*|7 |6*|    |1 |2*|3 |4*|    |2 |3*|4 |5*|    |8 |7*|6 |5*|
+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+
|2 |3*|4 |5*|    |8 |7*|6 |5*|    |1 |8*|7 |6*|    |1 |2*|3 |4*|
+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+

+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+
|2*|1 |8*|7 |    |8*|1 |2*|3 |    |3*|4 |5*|6 |    |7*|6 |5*|4 |
+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+
|3*|4 |5*|6 |    |7*|6 |5*|4 |    |2*|1 |8*|7 |    |8*|1 |2*|3 |
+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+

+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+
|3 |2*|1 |8*|    |7 |8*|1 |2*|    |4 |5*|6 |7*|    |6 |5*|4 |3*|
+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+
|4 |5*|6 |7*|    |6 |5*|4 |3*|    |3 |2*|1 |8*|    |7 |8*|1 |2*|
+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+

+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+
|4*|3 |2*|1 |    |6*|7 |8*|1 |    |5*|6 |7*|8 |    |5*|4 |3*|2 |
+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+
|5*|6 |7*|8 |    |5*|4 |3*|2 |    |4*|3 |2*|1 |    |6*|7 |8*|1 |
+--+--+--+--+    +--+--+--+--+    +--+--+--+--+    +--+--+--+--+

>>16 NORTHWEST POSITIONS

   +--+--+       +--+--+       +--+--+       +--+--+
   |1 |8*|       |1 |2*|       |8*|1 |       |2*|1 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|3*|2 |7*|    |7*|8 |3*|    |6 |7*|2 |    |4 |3*|8 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|4*|5 |6*|    |6*|5 |4*|    |5 |4*|3 |    |5 |6*|7 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+

   +--+--+       +--+--+       +--+--+       +--+--+
   |7*|6 |       |3*|4 |       |2 |3*|       |8 |7*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|1 |8*|5 |    |1 |2*|5 |    |8*|1 |4*|    |2*|1 |6*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|2 |3*|4 |    |8 |7*|6 |    |7*|6 |5*|    |3*|4 |5*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+

   +--+--+       +--+--+       +--+--+       +--+--+
   |7*|8 |       |3*|2 |       |6*|5 |       |4*|5 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|5 |6*|1 |    |5 |4*|1 |    |8 |7*|4 |    |2 |3*|6 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|4 |3*|2 |    |6 |7*|8 |    |1 |2*|3 |    |1 |8*|7 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+

   +--+--+       +--+--+       +--+--+       +--+--+
   |5 |6*|       |5 |4*|       |6*|7 |       |4*|3 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|3*|4 |7*|    |7*|6 |3*|    |4 |5*|8 |    |6 |5*|2 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|2*|1 |8*|    |8*|1 |2*|    |3 |2*|1 |    |7 |8*|1 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+

>>16 NORTHEAST POSITIONS

+--+--+       +--+--+       +--+--+       +--+--+
|1 |8*|       |1 |2*|       |8*|1 |       |2*|1 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|2 |7*|6 |    |8 |3*|4 |    |7*|2 |3*|    |3*|8 |7*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|3 |4*|5 |    |7 |6*|5 |    |6*|5 |4*|    |4*|5 |6*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+

+--+--+       +--+--+       +--+--+       +--+--+
|8 |7*|       |2 |3*|       |7*|8 |       |3*|2 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|1 |6*|5 |    |1 |4*|5 |    |6*|1 |2*|    |4*|1 |8*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|2 |3*|4 |    |8 |7*|6 |    |5*|4 |3*|    |5*|6 |7*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+

+--+--+       +--+--+       +--+--+       +--+--+
|6 |7*|       |4 |3*|       |3 |4*|       |7 |6*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|5 |8*|1 |    |5 |2*|1 |    |2 |5*|6 |    |8 |5*|4 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|4 |3*|2 |    |6 |7*|8 |    |1 |8*|7 |    |1 |2*|3 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+

+--+--+       +--+--+       +--+--+       +--+--+
|6*|5 |       |4*|5 |       |5 |6*|       |5 |4*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|7*|4 |3*|    |3*|6 |7*|    |4 |7*|8 |    |6 |3*|2 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|8*|1 |2*|    |2*|1 |8*|    |3 |2*|1 |    |7 |8*|1 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+

>>16 SOUTHWEST POSITIONS

+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|1 |2*|3 |    |1 |8*|7 |    |8*|1 |2*|    |2*|1 |8*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|8 |7*|4 |    |2 |3*|6 |    |7*|6 |3*|    |3*|4 |7*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
   |6*|5 |       |4*|5 |       |5 |4*|       |5 |6*|
   +--+--+       +--+--+       +--+--+       +--+--+

+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|3 |2*|1 |    |7 |8*|1 |    |2 |3*|4 |    |8 |7*|6 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|4 |5*|8 |    |6 |5*|2 |    |1 |8*|5 |    |1 |2*|5 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
   |6*|7 |       |4*|3 |       |7*|6 |       |3*|4 |
   +--+--+       +--+--+       +--+--+       +--+--+

+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|3*|4 |5*|    |7*|6 |5*|    |4 |3*|2 |    |6 |7*|8 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|2*|1 |6*|    |8*|1 |4*|    |5 |6*|1 |    |5 |4*|1 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
   |8 |7*|       |2 |3*|       |7*|8 |       |3*|2 |
   +--+--+       +--+--+       +--+--+       +--+--+

+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|4*|5 |6*|    |6*|5 |4*|    |5 |4*|3 |    |5 |6*|7 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|3*|2 |7*|    |7*|8 |3*|    |6 |7*|2 |    |4 |3*|8 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
   |1 |8*|       |1 |2*|       |8*|1 |       |2*|1 |
   +--+--+       +--+--+       +--+--+       +--+--+

>>16 SOUTHEAST POSITIONS

+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|1 |2*|3 |    |1 |8*|7 |    |8*|1 |2*|    |2*|1 |8*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|8 |5*|4 |    |2 |5*|6 |    |7*|4 |3*|    |3*|6 |7*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|7 |6*|       |3 |4*|       |6*|5 |       |4*|5 |
+--+--+       +--+--+       +--+--+       +--+--+

+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|3 |2*|1 |    |7 |8*|1 |    |2 |3*|4 |    |8 |7*|6 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|4 |7*|8 |    |6 |3*|2 |    |1 |6*|5 |    |1 |4*|5 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|5 |6*|       |5 |4*|       |8 |7*|       |2 |3*|
+--+--+       +--+--+       +--+--+       +--+--+

+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|5*|4 |3*|    |5*|6 |7*|    |4 |3*|2 |    |6 |7*|8 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|6*|1 |2*|    |4*|1 |8*|    |5 |8*|1 |    |5 |2*|1 |
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|7*|8 |       |3*|2 |       |6 |7*|       |4 |3*|
+--+--+       +--+--+       +--+--+       +--+--+

+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|3 |4*|5 |    |7 |6*|5 |    |6*|5 |4*|    |4*|5 |6*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|2 |7*|6 |    |8 |3*|4 |    |7*|2 |3*|    |3*|8 |7*|
+--+--+--+    +--+--+--+    +--+--+--+    +--+--+--+
|1 |8*|       |1 |2*|       |8*|1 |       |2*|1 |
+--+--+       +--+--+       +--+--+       +--+--+

>>END OF CATALOG