Roland Silver <rollos@starband.net> writes:

... I published a paper on the automorphisms the board (permutations of the cells which preserve 4-in-a-row) some years ago (Amer. Math. Monthly; Vol. 74, No. 3 (March 1967), pp. 247-254). Aside from the obvious subgroup of 48 rotations and reflections, there are two other independent automorphisms. One scrambles the board by interchanging each of the 3 parallel pairs of inner planes, while the other inverts the board by interchanging each of the 3 pairs consisting of an inner plane and its parallel outer neighbor.

The whole group has 192 elements.

I enjoyed that paper quite a bit. I especially appreciated the work you did to show that there were no _other_ symmetries. Later on, I was working with "Rubik's Revenge", the 4^3 Rubik's cube. I think it was David Singmaster who noticed that it had some extra symmetries, and I realized with amazement that they were the same as the extra symmetries of 4^3 tic-tac-toe. (These symmetries only work if you consider the puzzle to consist of an array of cubes all the way through, where the inner cubes have to be solved as well as the outer ones.) These symmetries--inverting some layers and permuting the noncentral layers--extend to larger sizes and dimensions of Rubik's cubes and tic-tac-toe boards. I suspect these are all there are, no matter how big you get, but I haven't a proof. I wonder if these groups have any other applications? Dan Hoey@AIC.NRL.Navy.Mil