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I hacked up a quick simulator for the 3x3x1 Rubik's Square today and I noticed that the number of positions at each level was the same for the ( U2, D2, F2, B2 ) subgroup of the normal 3x3x3 cube.

Analysis of ( U2, D2, F2, B2 )
------------------------------

Level   Number of Positions
                
   0             1
   1             4
   2            10
   3            24
   4            53
   5            64
   6            31
   7             4
   8             1
               ---
               192

Everyone agree? I'm not sure if this has been pointed out before.