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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