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I wish to propose some terminology and definitions to make certain

concepts a bit more precise. For example, when we are talking about

"corners only", it is not always clear whether we are talking about

"corners with centers without edges" or "corners without centers

without edges". In this note, I have tried to be consistent with

previous usage on the list, but I welcome any historical corrections

that might be deemed necessary.

Let G be the standard cube group for the 3x3x3 cube, and let |G| be the size G. Hence, we have |G| = (8!(3^8)/3 * 12!(2^12)/2) / 2, which is the famous 4.3 * 10^19.

Let GC be the corners with centers without edges group for the 3x3x3

cube, and let |GC| be the size of GC. Hence, we have

|GC| = 8!(3^8)/3. (I welcome a suggestion other than "GC" as the

name for this group. I did not find one in the archives.) This group

could be modeled by removing the edge labels from a standard

3x3x3 cube.

Let GE be the edges with centers without corners group for the 3x3x3

cube, and let |GE| be the size of GE. Hence, we have

|GE| = 12!(2^12)/2. (As before, I welcome a suggestion other than

"GE" for the name for this group.) This group could be modeled by

removing the corner labels from a standard 3x3x3 cube.

Note that |G| = |GC| * |GE| / 2.

Let G\C be the corners with edges without centers group. I intend

for the notation to indicate G reduced by C, where C is the rotation

group for the cube. It should be the case that |G\C| = |G| / 24,

but I want to return to this question a little later. This group

could be modeled by removing the center labels from a standard

3x3x3 cube.

Let GC\C be the corners without edges without centers group. This is

the 2x2x2 cube. We should have |GC\C| = |GC| / 24, but again I want

to return to this question a little later. In addition to being the

2x2x2 cube, this group could be modeled by removing the center and

edge labels from a standard 3x3x3 cube.

Let GE\C be the edges without corners without centers group. We

should have |GE\C| = |GE| / 24, but again I want to return to this

question a little later. This group could be modeled by removing

the center and corner labels from a standard 3x3x3 cube.

Let G\M be the set of M-conjugate classes for G. In this case,

|G\M| is approximately 48 times smaller than |G|. I believe that

when Dan Hoey asked in 1984 the question "how big is G, really?",

that he was really asking how big is G\M, and that he was asking

for the approximation to be resolved to an exact number.

Let GC\M be the set of M-conjugate classes for GC. In this case,

|GC\M| is approximately 48 times smaller than |GC|.

Let GE\M be the set of M-conjugate classes for GE. In this case,

|GE\M| is approximately 48 times smaller than |GE|.

Recall that B is the function which calculates the canonical form

for a cube under the composed operations of M-conjugation plus

rotation. My programs calculate equivalence classes under B.

Let G\B be the set of B-classes for G. Let GC\B be the set of B-classes

for GC. Let GE\B be the set of B-classes for GE. So far, my programs

have built complete search trees for GC\B and GE\B.

Let Gx denote any of G, GC, and GE. Then, we have Gx\B=(Gx\C)\M=(Gx\M)\C.

In English, we can decompose B into a multiplication by C and M (in

either order).

Also, note that Gx\C=(Gx\C)\C=((Gx\C)\C)\C=.... Similarly,

(G\M)\C=((G\M)\C)\C=.... In English, having reduced once by C, we can

reduce again by C as many times as we wish, but we simply get the same

set back again each time.

This notation can help us address the question of whether B actually

accomplishes a "times 48" or a "times 1152" reduction in the size of

the cube. If we are dealing with Gx, then Gx\B is a "times

1152" reduction. However, information is lost. For example, consider

GC and GC\B. GC is "corners plus centers", and B-reduction of GC

removes the centers and calculates M-conjugates of the corners.

But you really don't have the same problem any more because the centers

are gone. If on the other hand we are dealing with Gx\C, then (Gx\C)\B

is a "times 48" reduction. All we have really done is calculate

M-conjugates. The reduction by the C that is composed into B is duplicate

effort which accomplishes nothing.

I have come to realize that my program for the 2x2x2 actually models

GC (corners with centers without edges) rather than GC\C (corners

without centers without edges). My program does not explicitly encode

the centers. However, it encodes all eight corner cubies, and when

it makes qturns, any of the eight cubies can move. Hence, rotational

information is encoded, even if the centers themselves are not explicitly

encoded. If I wanted to model GC\C, I would have had to either model

only seven of the cubies, or else modeled all eight but moved only seven

of them. Since what I really wanted was (GC\C)\M, and since what I had

was GC, I had to invent this funny B thing, where GC\B=(GC\C)\M. If I

had been clever enough to model GC\C in the first place, I never would

have had to invent B. Similar comments apply to my model for the edges.

To convince yourself that eight corner cubies model GC and seven corner

cubies model GC\C, just think about calculating |GC| and |GC\C|.

For |GC|, there are eight ways to pick the first cubie, seven ways to

pick the second cubie, and so forth yielding the familiar

|GC|=8!*(3^8)/3. For |GC\C|, we let one of the cubies be fixed,

then there are seven ways to pick the second cubie, and so forth yielding

|GC\C|=7!*(3^7)/3, and |GC| = |GC\C| * 24. Hence, the "corners of the

3x3x3" problem is 24 times larger than the "2x2x2" problem.

I will discuss the "times 24" reduction that is accomplished by

reducing by C in a followup note.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU

If you don't have time to do it right today, what makes you think you are

going to have time to do it over again tomorrow?