On Tue, 4 Jun 1996, Jim Mahoney wrote:

It's going to take a while to absorb your whole note, but I do have a
couple of quick comments/questions.

Next, I will define any complete set of cubies that can move into each
other's position as an "orbit." (This name is at least suggestive of
the group theory notion of a closed sequence of elements.) For
example, the 8 corner cubies on the 3x3x3 Cube form one orbit since
any one of those cubies can be put in any of those eight positions.
Likewise, the 12 edge cubies on the 3x3x3 form another orbit.

This has been discussed before on Cube-Lovers, but I am still puzzled or
curious about the usage of the word "orbit". Your definition is
consistent with the usage advocated by Martin Schoenert on Cube-Lovers.
For example, Martin talked about the corner orbit, the edge orbit, and the
face center orbit of the 3x3x3. (I suppose for completeness, we should
include in this list of orbits the orbit for the invisible center of the
whole 3x3x3 cube.)

David Singmaster, on the other hand, has always talked about the twelve
orbits of the constructable group of the 3x3x3, where orbits are defined
in terms of twists, flips, and parity. Depending on what you mean by
"closed sequence of elements", your definition may be consistent with
Singmaster's usage as well. That is, Singmaster's orbits are certainly
closed. However, Martin says that Singmaster's orbits should be called
cosets.

Secondly, if my understanding of your model is correct, you are treating
positions as distinct which cannot be distinguished with normal coloring
of a physical cube (even an imaginary physical cube for large N). The
issue appears as early as the 4x4x4, and persists for larger values of N.
I don't necessarily disagree with your treatment. Indeed, it makes the
cube theory tenable. Otherwise, your model tends to become a coset model
rather than a group model. But I wondered if my understanding of your
model is correct?

There are several implications of how you treat visibly indistinguishable
positions. For example, it impacts your counts of how many positions
there are. For another example, it impacts your solutions (e.g.,
"invisible" incorrect parity on the 4x4x4. "Invisible" bad parity can
also occur on the 3x3x3 if you remove the face center color tabs. A slice
move will give the edges and corners opposite parity that is not visible.)
Perhaps you could discuss these issues with respect to your model.

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Robert G. Bryan (Jerry Bryan)                jbryan@pstcc.cc.tn.us
Pellissippi State                            (423) 539-7127
10915 Hardin Valley Road                     (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990