Additive-multiplicative magic squares, 8th and 9th-order,
Smallest multiplicative magic squares, 8th and 9th-order,
Pandiagonal multiplicative magic squares, 8th and 9th-order.

After multiplicative magic squares of orders 3-4-56-7, and before multiplicative magic squares of orders >=10 and additive-multiplicative magic squares of orders >=10,  here are the most interesting results on orders 8-9.

Walter W. Horner (1894 - 1988)

In the 1950s, Walter W. Horner, an American teacher of mathematics, constructed the first known additive-multiplicative magic squares (also called addition-multiplication squares), later republished by Joseph S. Madachy and J. A. H. Hunter. When you multiply the integers in each row, column or diagonal, you get the same product P. When you add the integers in each row, column or diagonal, you get the same sum S.

 162 207 51 26 133 120 116 25 105 152 100 29 138 243 39 34 92 27 91 136 45 38 150 261 57 30 174 225 108 23 119 104 58 75 171 90 17 52 216 161 13 68 184 189 50 87 135 114 200 203 15 76 117 102 46 81 153 78 54 69 232 175 19 60

 200 87 95 42 99 1 46 108 170 14 44 10 184 81 85 150 261 19 138 243 17 50 116 190 56 33 5 57 125 232 9 7 66 68 230 54 4 70 22 51 115 216 171 25 174 153 23 162 76 250 58 3 35 88 145 152 75 11 6 63 270 34 92 110 2 28 135 136 69 29 114 225 27 102 207 290 38 100 55 8 21

Gakuho Abe, Japan, constructed some years later, a 9x9 additive-multiplicative magic square, but with a bigger P = 1,619,541,385,529,760,000. This square is also reported by Joseph S. Machady.

In November 2005, I constructed better 8x8 and 9x9 additive-multiplicative squares, "better" meaning with smaller constants. My smallest 8x8 and 9x9 products are respectively about 40 times and 2 times smaller than Horner's products mentioned above.

• Here are my best results optimizing P, S and Max nb for 8x8 additive-multiplicative magic squares:

•  222 66 225 63 5 7 68 104 1 35 52 136 198 74 189 75 132 296 21 175 9 15 78 34 45 3 102 26 148 264 25 147 51 117 10 6 200 84 259 33 168 100 231 37 39 153 2 30 91 17 8 20 42 150 99 333 50 126 111 297 119 13 40 4

 75 38 207 102 11 20 91 56 5 44 49 104 57 50 153 138 133 200 17 92 45 66 21 26 99 30 39 14 175 152 23 68 78 63 22 15 184 119 100 19 136 161 76 25 42 117 10 33 28 13 40 77 34 69 114 225 46 51 150 171 52 7 88 35
• And here are my best results optimizing P, S and Max nb for 9x9 additive-multiplicative magic squares:

•  38 150 248 10 7 65 44 153 69 4 63 39 22 102 184 190 25 155 110 17 115 76 225 93 2 42 104 186 152 50 13 5 70 207 33 68 117 3 28 138 88 34 31 95 250 23 55 170 279 57 100 78 8 14 200 62 114 35 130 1 51 92 99 21 52 9 136 46 66 125 310 19 85 230 11 75 124 171 56 26 6

 84 145 133 80 11 6 104 243 34 40 99 2 78 135 119 224 29 114 208 27 102 112 261 38 30 55 7 95 196 87 1 60 88 153 52 108 9 20 44 85 182 81 19 168 232 17 156 216 171 56 116 5 70 33 203 57 140 66 8 10 54 68 234 22 4 90 189 51 130 174 152 28 162 136 26 58 76 252 77 3 50

Thanks to Ed Pegg Jr, USA, the first to check the properties of my 4 squares above.

Smallest multiplicative magic squares, 8th and 9th-order

All the above additive-multiplicative magic squares are also -obviously- multiplicative magic squares. But if we try to optimize P and Max nb, without the need to have additive properties, then it is possible to construct better squares.

• In 2005, Luke Pebody, England, constructed the two following squares, with the smallest possible products P:

•  27 7 55 32 221 10 114 8 152 13 64 15 9 14 33 85 77 30 3 26 48 95 136 6 5 102 38 88 21 12 24 65 20 40 39 17 76 36 42 11 4 66 45 35 2 68 52 228 51 16 28 19 44 78 25 18 104 57 34 54 50 22 1 56

 119 8 40 44 36 95 9 42 65 72 20 77 2 35 104 190 34 27 70 81 75 91 16 136 4 19 22 13 88 1 64 63 45 25 84 323 3 26 102 15 76 55 98 90 32 114 50 28 30 17 18 33 52 56 10 51 48 133 110 14 39 12 60 80 49 57 54 130 24 68 11 5 66 38 78 85 6 7 96 100 21
• And I constructed the 2 following squares with bigger P, but smaller max nbs than Luke's examples.

•  119 80 84 55 3 152 18 13 6 95 63 52 34 20 42 88 33 112 40 17 91 36 114 5 26 9 57 8 66 70 140 68 72 39 1 38 56 77 85 120 14 22 136 60 45 78 4 133 76 7 65 54 160 51 11 28 100 102 44 98 19 2 104 27

 184 1 84 133 75 65 99 34 96 55 119 72 92 9 28 114 200 13 38 100 117 11 102 192 69 5 98 24 66 136 42 115 7 26 76 225 91 57 125 216 22 68 112 23 6 56 207 2 78 152 25 120 77 51 150 104 19 85 168 33 4 126 46 3 70 161 50 52 171 17 144 88 153 48 44 8 14 138 175 39 95
• Then in February 2013, Toshihiro Shirakawa improved on my results, with the 2 following squares having the smallest known Max Nb (and probably the smallest possible).

•  117 1 48 66 5 112 105 60 20 96 25 56 33 13 63 18 7 77 3 72 104 24 100 45 108 65 49 10 30 9 64 22 16 50 27 4 70 42 99 52 21 36 78 90 32 110 2 35 40 28 44 91 81 75 6 8 55 54 80 15 14 12 26 84

 90 48 153 66 7 128 1 65 105 140 81 3 50 96 28 26 51 44 16 85 14 52 25 99 144 126 6 9 78 20 68 70 5 132 112 54 119 100 108 21 88 10 36 8 39 18 22 30 72 34 13 147 120 40 11 42 91 45 27 136 80 15 32 104 2 110 84 24 63 75 12 102 60 56 64 4 117 135 17 33 35

Pandiagonal multiplicative magic squares, 8th and 9th-order

• In May 2006, I constructed the following pandiagonal 8x8 and 9x9 multiplicative squares. The 8x8 square is pandiagonal, but also a most-perfect square, because all its 2x2 subsquares (as for example the green one) have the same product P'. This 8x8 square is also a Franklin square: its bent diagonals (as for example the blue one) have the same product P than all the other lines and diagonals.
The 9x9 square has all its 3x3 subsquares (as for example the green one) with the same product P as the lines and diagonals.

•  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

 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
• The two above squares are not the best possible pandiagonal squares. One year later, in May-June 2007, I constructed better squares having smaller P and Max nb with Jaroslaw Wroblewski, Poland. They lose some of the above characteristics (no longer most-perfect, or 3x3), but they became the new best known 8x8 and 9x9 pandiagonal magic squares.

•  1 78 88 45 15 182 462 108 42 216 13 33 8 90 195 77 135 7 546 264 9 3 104 110 44 10 270 91 231 24 18 39 22 27 4 130 330 63 21 312 273 132 2 54 52 55 30 126 390 154 189 12 26 66 36 5 72 65 165 14 378 156 11 6

 12 54 77 50 24 378 110 160 10 528 120 18 7 165 240 126 80 42 270 176 40 6 189 55 63 5 264 420 90 16 132 60 44 20 21 135 88 140 30 432 300 144 4 66 210 45 8 462 216 154 100 48 108 22 70 15 231 150 72 14 330 480 36 2

 1 176 252 10 264 315 12 1760 378 140 432 11 14 288 110 21 360 132 40 924 180 48 77 18 32 770 27 630 3 440 756 20 528 63 2 352 22 28 720 33 35 864 220 42 72 54 8 154 36 80 231 45 96 1540 1056 1260 6 88 126 4 880 189 5 216 55 84 1440 66 7 144 44 70 308 90 24 385 108 160 462 9 16

Ten years later, in September 2017, Elbert Krison improved our square, moving its cells in order to obtain 3x3 subsquares:

 88 16 288 33 6 108 385 70 1260 231 42 756 440 80 1440 11 2 36 55 10 180 77 14 252 264 48 864 144 352 8 54 132 3 630 1540 35 378 924 21 720 1760 40 18 44 1 90 220 5 126 308 7 432 1056 24 32 72 176 12 27 66 140 315 770 84 189 462 160 360 880 4 9 22 20 45 110 28 63 154 96 216 528

This Max nb is no more the record. Also in 2017, Elbert constructed another 9x9 square with the new smallest known Max nb = 1365.