Given a 3x3 square with 9 distinct entries, can all permutations of the elememts in the square be reached when the only legal operation is to rotate a 2x2 subsquare 90 deg clockwise? (A rotation on the same subsquare may be done multiple times.) If not how many positions are attainable?

Example, rotating the upper left 2x2 square.
1 2 3    4 1 3
4 5 6 -> 5 2 6
7 8 9    7 8 9

(In reply to solution by Charlie)

I did this with pencil and paper, trying to show just two cases: swapping the center with an edge, and a corner with an edge.

Center with Edge:

123  413  413  153  136  136  123  123  123  123
456  526  756  476  457  527  576  569  459  465
789  789  829  829  829  489  489  478  768  789

Corner with Adjacent Edge:

123  413  413  243  214  164  143
456  526  276  816  863  283  268
789  789  579  579  579  579  579

213  213  213  213  213  213  213
648  568  546  756  745  486  456
579  749  798  978  986  796  789

From these two cases (and their reflections and rotations), swapping any two single pieces is possible; and it is easily shown from there that any configuration is possible to return to the "pristine" state.

Here's how to swap any pair of digits, using only previously determined moves.

Corner to center:

123  213  253  523  523
456  456  416  416  416
789  789  789  789  789

Adjacent corners:

123  213  231  321
456  456  456  456
789  789  789  789

Opposite Corners:

123  321  329  923
456  456  456  456
789  789  781  781

Adjacent Edges:

123  153  153  163
456  426  462  452
789  789  789  789

Opposite Edges:

123  153  153  183
456  426  486  456
789  789  729  729

Corner with Opposite Edge:

123  523  523  623
456  416  461  451
789  789  789  789

Edited on November 10, 2004, 11:38 am

Posted by DJ on 2004-11-10 11:37:10