Date: 8 Dec 1980 17:03 PST
From: McKeeman.PA at PARC-MAXC
Subject: A Proposed Definition of Symmetry
To: cube-lovers at MIT-MC

The discussion of local maxima for the Q measure of distance led to an informal
use of symmetry.  It is not clear to me just what symmetry is needed to carry
through the maxima argument but I suggest the following is sufficient (although
perhaps too restrictive).

Let C by the rotation group of the cube (closure of IJK: order 24)
Let G be Rubik's group (closure of UDRLFB: order 10^19 or so)

Both groups can be represented as a permutation group on [0, 1, ...53] for some
arbitrary numbering of the 54 faces.  We can also use the names UDRLFB for the
six colors; where the association is made once and for all for any given physical
puzzle.  Like U=red, F=blue, etc.).

The elements of g are 1-1 with the observable configurations of the standard
cube; and in fact are the recipes to reach the configurations from "home".  g' is
the "solution" that returns g to home.

The elements of G*C are also 1-1 with the observable configurations except now
the correspondence must also take into account the observed orientation of the
cube.

Each g in G is represented by a permutation of the cubelet faces.  Each face in g
is a fixed color.

For color X, let X[g] be the set of faces of g colored X.  |X[g]| = 9.
Let Coloring[g] = {U[g], D[g], R[g], L[g], F[g], B[g]}.

Then g is totally symmetric if for all c in C, Coloring[gc] = Coloring[g].

----

It is true that "home" and UUDDRRLLFFBB are totally symmetric by this
definition.  "home" is a minimum (special case).  UUDDRRLLFFBB is a local
maximum.

Questions:

Is there a simpler equivalent definition?

How many totally symmetric configurations are there?

Is there a less restrictive definition that guarantees local maxima?