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The first problem for the 4x4x4 cube is in eliminating

positions that arise from whole-cube moves. This was done with the

3x3x3 by keeping the face-center positions fixed, but there are no

face centers on the 4x4x4--or there are, but they don't maintain a

fixed orientation relative to each other. I standardize the spatial

orientation by keeping the DBR corner in a fixed position and

orientation. A move then consists of twisting one, two, or three

layers parallel to the U, F, or L face. Thus F3' is equivalent to

twisting the B face. I will refer to the number of layers twisted

as the "depth" of the move.

Following David C. Plummer's notation (31 DEC 1980 1210-EST),

organize each face of the cube as

C L R C

R X X L

L X X R

C R L C.

I will assume familiarity with David Vanderschel's analysis of the

3x3x3 case, which was presented in his message of 6 August 1980.

"C" faces act as they do in the 3x3x3 case, except that one

of them does not move. Corner Orientation Parity (COP) is preserved

and Corner Permutation Parity (CPP) changed by every quarter-twist.

Depending on the depth, a quarter twist can permute the "L"

faces in an odd or an even permutation. Also, "L" faces do not

change orientation (or move to "R" positions). Every "R" face is

determined by the "L" face (on an adjacent side of the cube) with

which it shares a cubie. Thus the arguments for EOP and EPP do not

apply.

Every quarter-twist is an odd permutation of the "X" faces:

either one, three, or five four-cycles, depending on the depth.

Letting XPP be the permutation parity of the "X" faces, the Total

Permutation Parity TPP=XPP+CPP (mod 2) is preserved by every

quarter-twist.

Thus the 4x4x4 cube group has at least six orbits,

according to COP (mod 3) and TPP (mod 2). The basic upper bound of

7! Corner Permutations

3^7 Corner Orientations

24! L Permutations (which determine the R permutations), and

24! X Permutations,

divided by six, yields an upper bound (of about 7.072*10^53). I have

run Furst's algorithm on the problem, and my program claims that all

these positions are reachable.

To calculate the number of reachable color patterns, note

that there are 4! permutations of each quadruple of "X" faces which

are indistinguishable. However, the TPP constrains the XPP so as to

reduce this by a factor of two. Dividing 7.072*10^53 by (4!)^6/2

yields 7.401*10^45.

[At this point, you may find it instructive to view the

message before last, which analyzes the 5x5x5 cube in the context

of this message and the one immediately preceding. I regret the

accidental disorder. These three are all for now, although I have

results on tetrahedra, octahedra, and a dodecahedron which I am in

the process of writing up.]