   Date: Wed, 24 Jun 87 02:40:53 -0400 (EDT)   From: Dan Hoey <hoey@nrl-aic.ARPA >
~~~ ~~~ Subject: Groups of the larger cubes

Last year Rodney Hoffman cited an article by J. A. Eidswick (in the
March 1986 Math Monthly) that develops a general approach to analyzing
several magic polyhedra. Did anyone else go read this one? Of
particular interest is Eidswick's analysis of the larger three-
dimensional cubes. The article shows that the only constraints on
these cubes are the permutation parity constraints implicit in the
generators and the corner and edge orientation constraints we already
know about. Eidswick shows that this even holds for the ``theoretical
invisible group'', where we imagine that the interior of the magic
N-cube is a magic (N-2)-cube that must be solved simultaneously. The
solution method he presents is to solve the parity problems by applying
zero or one qtw at each of the floor(N/2) depths, then to work with
commutators (aka mono-ops) to solve the rest of the cube, piece by
piece.

As a supplement to that article, here are the number of positions
G[t](N) of the N^3 magic cube, where t, a subset of {s,m,i},
indicates the set of traits we find interesting:

s (for N odd) indicates that are working in the Supergroup, and
so take account of twists of the face centers.
m (for N > 3) indicates that the pieces are marked so that we
take account of the permutation of the identically-colored
pieces on a face.
i (for N > 3) indicates that we are working in the theoretical
invisible group, and solve the pieces on the interior of the
cube as well as the exterior. I will assume that the M and S
traits apply to the interior pieces as if they were on the
exterior of a smaller cube.

A formula for the number of positions is

```           2^A (8!/2 3^7)^B (12!/2 2^11)^C (4^6/2)^D (24!/2)^E
G[t](N) =  ---------------------------------------------------
24^F (24^6/2)^G
```

The following table gives the values of parameters A-G, depending on
the traits, and on whether N is even or odd.

```Parameter   Traits  (N odd)              (N even)
```
```(Parity)        A =         (N-1)/2              N/2

(Corners)       B = i       (N-1)/2              N/2
~i      1                    1

(Edge centers)  C = i       (N-1)/2              0
~i      1                    0

(Face centers)  D = ~s      0                    0
s,i     (N-1)/2              0
s,~i    1                    0

(Other cubies)  E = i       (N+4)(N-1)(N-3)/24   N(N^2-4)/24
~i      (N+1)(N-3)/4         N(N-2)/4

(Whole-cube)    F =         0                    1

(Color cosets)  G = m       0                    0
~m,i    (N^2-1)(N-3)/24      N(N-1)(N-2)/24
~m,~i   (N-1)(N-3)/4         (N-2)^2/4
```

In any case, the size of the group is exponential in a polynomial in N;
the polynomial is cubic if trait "i" is present and quadratic otherwise.
Here is a table of numeric approximations for cubes up to 10^3.

```              Traits excluding s
N     {}        {m}       {i}      {m,i}
2  3.674e6   3.674e6   3.674e6   3.674e6
3  4.325e19  4.325e19  4.325e19  4.325e19
4  7.401e45  7.072e53  3.263e53  3.118e61
5  2.829e74  2.583e90  6.117e93  5.585e109
6  1.572e116 1.310e148 3.077e170 2.451e210
7  1.950e160 1.484e208 2.982e253 2.072e317
8  3.517e217 2.335e289 3.247e388 1.717e500
9  1.417e277 8.208e372 5.283e529 2.126e689
10  8.298e349 4.007e477 4.041e738 1.032e978
```
```             Traits including s
N     {s}      {s,m}     {s,i}    {s,m,i}
3  8.858e22  8.858e22  8.858e22  8.858e22
5  5.793e77  5.289e93  2.566e100 2.343e116
7  3.994e163 3.039e211 2.562e263 1.780e327
9  2.902e280 1.681e376 9.293e542 3.740e702
```

Enough, then, of what are essentially Eidswick's results. In my next
message, I plan to produce lower bounds for solving these cubes.

Dan     