hihi, all - i think i proved something once long ago that might help i was thinking about the set of possible configurations in the 15-puzzle, and in some generalizations of it to other ``outlines'' (i.e., shapes other than a large surrounding square) - the specific problem was to recognize starting points that could not reach the expected solution pattern i argued successfully (to myself at least), that in any arrangement of unit squares that was connected by paths of width at least 2 (that is, if a 2x2 square can be moved around inside the outline), then any even permutation of the original unit squares is possible that leaves the initial empty square in the same place (the fact that all and only even permutations are possible was given as the solution of the reachability problem for the square outline, and has apparently been known for some time) - i don't think my argument depended on the outline being simply connected, just ``thick enough'' to have the 2x2 squares fit everywhere (i.e., no narrow necks) this connects the problem to a generator word counting problem in a known (though large, of course) group with a determinable set of generators (i think it was 3-cycles on any 3 unit squares in any 2x2 square, but i don't exactly remember how or if i dealt with the cosets, that is, the different locations of the empty square - i know i did not consider the word lengths at all) there is also a question of what constitutes a move - whether one move is sliding just one of the unit squares into the adjacent empty square, or if an entire row or column (or part of one) can slide as a single move (i prefer the former choice, but i've seen others that prefer the latter, since it is more like how we actually make the moves in the game) more later, cal