Moves and thoughts about the Megaminx ------------------------------------- The megaminx is a platonic dodecahedron with a rubik type mechanism. I.e. it has 12 sides which rotate. There are two types: The Hungarian SuperNova & The Megaminx manufactured by Tomy. The edge pieces are a bit thicker on the Megaminx than the SuperNova. The megaminx has a slice group, analagous to the cube slice group. All the possible spot patterns are in the megaminx's slice group, e.g. the 10 spot and the 12 spot patterns. With process M1 we may easily generate any spot pattern, although there is much room for improvement. During slice moves 10 corners and 10 edges move and 10 corners and 20 edges remain stationary. The slice group of the megaminx is generated by turning the faces opposite to each other in the opposite direction (i.e. opposite looking at a face head-on!) We aren't going to be exhaustively searching the slice group of the megaminx any time soon as GAP shows it contains 9,627,755,206,121,277,812,101,663,948,800,000 elements or approximately 9.6 * 10^33 The GAP has also shown that: Size (megaminx) = Size (DerivedSubgroup (megaminx)), so that the anti-slice group of the megaminx is isomorphic to the whole megaminx group. After doing a Monte Carlo analysis of the various orders of the megaminx group, the largest order known is length 720,720. Other known orders include: 540,540 471,240 360,360 353,430 240,240 102,102 10,080 5 In the case of process M1, L is opposite to R, not just separated by a face F, as in processes M2, M3a and M3b. My original diagram is rather limited, but it does illustrate the idea of L & R separated by F only (as opposed to real opposites but I have no satisfactory notation). /\ / \ / \ \ U / L \ / R \____/ F Moves for the Magic Dodecahedron (Megaminx) ------------------------------------------- C_U = Rotate entire dodecahedron clockwise via the U face suffix notation: f = face turns u = unit turns M1a 10 spot (shift 1) (L+ R- C_U-)^36 (slice group) (72f) M1b 10 spot (shift 2) (L++ R-- C_U+)^48 (slice group) (96f) M2 3 cycle of edges (uf, lf, rf) R+ F+ U+ F- U- R- L- U- F- U+ F+ L+ (12f) M3a 2 flip L-- R++ F+ U- R+ U+ L++ R++ U+ (18f, 26u) R-- L-- U- R- U+ F- R-- L++ U- M3a 2 flip (ul, uf) R- F- U+ L- U- L+ F+ R+ (16f, 16u) L+ F+ U- R+ U+ R- F- L-