   Date: Sun, 01 Feb 81 06:51:00 -0500 (EST)   From: Dan Hoey <Hoey@CMU-10A >
~~~ ~~~ Subject: Analysis of the 4x4x4 cube

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.]     