Back after a sunny week-end. Some remarks on the orders 6, 8, 9 and 12: ----------------- 6 As you know, my pandiagonal example has all its 2x2 sub-squares with the same product. I had forgotten to mention that it has also all its 3x3 sub-squares with the same product!! Both 2x2 and 3x3, a very pretty square. ----------------- 8 Your analysis of bit planes is interesting, Michael. Unfortunately, as remarked by Franklin, your square does not use distinct integers. However, you are right, I am not sure that my 8x8 square uses the smallest possible integers and product: it is currently my best possible, but I will be happy if somebody can succeed to use smaller characteristics. About "Franklin", and using the same set of integers than my previous example, here is a new pandiagonal square with a supplemental nice property: all its bent diagonals are magic. Here is the first Benjamin FRANKLIN's multiplicative square: 1 1080 42 1260 3 360 14 3780 378 140 9 120 126 420 27 40 180 6 7560 7 540 2 2520 21 840 63 20 54 280 189 60 18 36 30 1512 35 108 10 504 105 168 315 4 270 56 945 12 90 5 216 210 252 15 72 70 756 1890 28 45 24 630 84 135 8 ----------------- 9 Michael, here is my pandiagonal example that you can analyse with bit planes. Again 2^ * 3^ * 5^ * 7^ integers: 28 350 35 1764 22050 2205 12 150 15 45 36 450 105 84 1050 245 196 2450 7350 735 588 50 5 4 3150 315 252 70 7 700 4410 441 44100 30 3 300 900 90 9 2100 210 21 4900 490 49 147 14700 1470 1 100 10 63 6300 630 175 140 14 11025 8820 882 75 60 6 18 225 180 42 525 420 98 1225 980 2940 294 3675 20 2 25 1260 126 1575 All its 3x3 sub-squares have the same product. As all my examples, I have tried to use the smallest entries and the smallest product. Of course, not sure that they are the smallest possible! Again, I will be happy if somebody can improve it. ----------------- 12 Michael, your example has a magic product 1.06 E+56, with 2^ * 3^ integers. My best square has a better magic product 1.86 E+30, with 2^ * 3^ * 5^ * 7^ * 11^ integers. I send you a direct message with this sample. I do not want to bother all [math-fun] people with too big squares... But if somebody else is interested by this 12x12 sample, send me a direct message. Christian.