Christian Boyer wrote:
Excellent analysis, Michael! And you will be credited of this nice 10x10 pandiag. multiplic. square in my future website update.
Glad you like it. Was the construction related to how you found the 6x6 that you mailed out earlier, or did you come across it through some more general search? Since the construction is so straightforward, here is a packaged-up version of it for Mathematica. (I don't speak mma fluently; experts, feel free to improve on my code.) Copy[n_][x_List] := Join @@ Table[x,{n}] B[B0_] := B0 - Min[B0] T[B_] := Copy[ Length[B[[1]]] ][ Map[ Copy[Length[B]], B ] ] M[T_] := Table[2^T[[i,j]] 3^T[[j,i]], {i,Length[T]}, {j,Length[T]}] The operation B0 -> B is in there because it seems nicer to express the building block as a matrix B0 with distinct integer entries, possibly negative, and constant row and col sums (which might be the same, if you're lucky enough that they are both zero). These B0s are more symmetric-looking and their construction more transparent, I think. For example, you mentioned a 12x12 in your earlier email, which I'll guess is most-perfect in the usual 2x2 sense. Here's a 12x12 most-perfect in that all 3x3s have the same product: b0 = {{-6,3,-3,6},{2,1,-2,-1},{4,-4,5,-5}} M[T[B[b0]]] = {{1, 3359232, 472392, 4096, 6561, 30233088, 8, 26873856, 59049, 512, 52488, 241864704}, {5038848, 279936, 144, 629856, 559872, 1152, 314928, 69984, 2304, 2519424, 34992, 288}, {27648, 324, 362797056, 54, 82944, 708588, 55296, 162, 181398528, 108, 165888, 354294}, {531441, 124416, 24, 2176782336, 243, 1536, 4251528, 995328, 3, 272097792, 1944, 12288}, {256, 839808, 944784, 32, 1679616, 7558272, 16, 209952, 15116544, 128, 104976, 1889568}, {20155392, 8748, 18432, 39366, 2239488, 36, 40310784, 4374, 9216, 78732, 4478976, 18}, {27, 41472, 1417176, 110592, 81, 90699264, 216, 331776, 177147, 13824, 648, 725594112}, {136048896, 31104, 48, 17006112, 62208, 384, 8503056, 7776, 768, 68024448, 3888, 96}, {1024, 26244, 120932352, 2, 6718464, 236196, 2048, 13122, 60466176, 4, 13436928, 118098}, {19683, 1119744, 72, 80621568, 2187, 4608, 157464, 8957952, 9, 10077696, 17496, 36864}, {6912, 10368, 2834352, 864, 20736, 22674816, 432, 2592, 45349632, 3456, 1296, 5668704}, {544195584, 972, 6144, 1062882, 248832, 12, 1088391168, 486, 3072, 2125764, 497664, 6}} The magic product is 2^72 3^72, while each 3x3 multiplies to 2^54 3^54. I haven't looked at your 8x8 example; did you find it via some general 2/3/5/7-based construction? From the results last time we talked about multiplicative magic squares, I'm not at all surprised that this generally gives smaller values than a 2/3-based construction alone. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.