### Overlapping orbits, take 2

A week ago, I wrote a post about
something I called "overlapping orbits" and a puzzle based on
them.

It's been pointed out to me that my use of "orbit" is
using a technical term from group theory for something other than its
technical meaning. (The orbit of a thing, as I was using it there, is
the set of places it can go under zero-or-more repetitions of a single
move; thus, a thing has multiple orbits, one for each move. The orbit
of a thing, as used in group theory, is the set of places it can go
under any sequence of moves. Thus, for example, the orbit of a corner
facie of a cube is the set of all corner facicles, since you can put a
corner facie into any corner facicle.)

This post is to mention two things: (1) the above terminological
note and (2) it occurred to me that allowing the rotations to overlap
grid boundaries, or, to put it another way, doing the thing on a torus
instead of a rectangular section of the Euclidean plane, might be
interesting.

The way the program I mentioned last post was structured, that
extension is trivial, because the program just has a list of
permutations on nine objects and walks the graph induced by them. So
all I had to do was augment the list of eight permutations I had (four
rotations each way) by the ten more representing these
crossing-the-boundary rotations.

I've done that. Of course, the graph of reachable positions is more
strongly connected; the greatest distance from start is 7, with the
commonest distance being 5, with 6 a close runner-up (the position
counts, by distance, are 1, 18, 297, 4206, 43248, 164472, 137894,
12744).

Interestingly, a single swap of adjacent numbers is much shorter; it
can be done in three moves. The procedure my program finds includes
moves that overlap in ways that make it impossible to use that sequence
on the original 3x3 grid for any pair of numbers, but I think I will
try all possible pair-swaps in the original puzzle (well, reduced by
symmetry) to see how different the resulting swap procedures are.

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