Received: from nrl-aic.ARPA (TCP 3200200010) by AI.AI.MIT.EDU 24 Jun 87 03:18:32 EDT Return-Path: Received: Wed, 24 Jun 87 03:12:52 edt by nrl-aic.ARPA id AA18187 Date: 24 Jun 1987 02:40:53 EDT (Wed) From: Dan Hoey Subject: Groups of the larger cubes To: Cube-Lovers@AI.AI.MIT.EDU Message-Id: <551515254/hoey@nrl-aic> 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