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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].

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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?