Christian Boyer wrote:
Here is an example!
7776 3 3888 96 243 48 2 46656 4 1458 64 2916 2592 9 1296 288 81 144 486 192 972 6 15552 12 32 729 16 23328 1 11664 162 576 324 18 5184 36
All the rows, columns, diagonals AND broken diagonals give the same product:[...] And best, it is a "most-perfect" pandiagonal magic square, à la Kathleen Ollerenshaw. All 2x2 sub-squares have always the same product:[...]
This is a very pretty construction! I think in the same way you can generate pandiagonal multiplictive magic squares of any size not a prime power (and various sub-square sizes of most-perfect-ness). Here's my interpretation of your example: --------- a) Take the 2x3 matrix [ 5 0 4 ] [ 1 6 2 ] This has relatively prime side lengths, distinct positive integer entries, a constant row sum, and a (different, of course!) constant column sum. Call this building-block matrix B. b) Form a 6x6 matrix by tiling with 6 copies of B. Now you have a square with constant row and col sum (equal to 18, the sum of the entries in B) and the same constant sum along all broken diagonals, because each diagonal traces a (2,3)-torus knot on B. Also, any 2x2 subsquare is exactly 2 columns of B, so will sum to 2/3 the other sums. Call this tiled matrix T. c) Make a multiplicative magic square M = 2^T 3^transpose(T). (This is pointwise exponentiation and multiplication.) --------- It is vital that the sides of the building block B are relatively prime (this ensures the pandiagonality and the fact that M's entries are all distinct). Oh, and the most-magic-ness is that all squares of side length the smaller side of B will have the same product. So to make a 10x10 square with all those desirable properties, you could start with the 2x5 rectangle [ 0 11 2 9 8 ] [ 12 1 10 3 4 ] and perform the same construction, to get 1, 725594112, 9, 80621568, 6561, 4096, 177147, 36864, 19683, 26873856 1088391168, 6, 120932352, 54, 165888, 1062882, 6144, 118098, 55296, 162 4, 181398528, 36, 20155392, 26244, 1024, 708588, 9216, 78732, 6718464 272097792, 24, 30233088, 216, 41472, 4251528, 1536, 472392, 13824, 648 256, 2834352, 2304, 314928, 1679616, 16, 45349632, 144, 5038848, 104976 531441, 12288, 59049, 110592, 81, 2176782336, 3, 241864704, 27, 331776 2048, 354294, 18432, 39366, 13436928, 2, 362797056, 18, 40310784, 13122 2125764, 3072, 236196, 27648, 324, 544195584, 12, 60466176, 108, 82944 512, 1417176, 4608, 157464, 3359232, 8, 90699264, 72, 10077696, 52488 136048896, 48, 15116544, 432, 20736, 8503056, 768, 944784, 6912, 1296 Every row/col/diag multiplies to 2^60 3^60 = 48873677980689257489322752273774603865660850176 and every 2x2 square multiplies to 2^24 3^24 = 4738381338321616896. For a 20x20 example you'd need a 4x5 B and every 4x4 block in the final square would have the same product. For an example of size pqr, like 30, you could start with a B of sizes 2x15, 3x10, or 5x6, and you'd get different most-magic subsquare sizes for each. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.