Additive-multiplicative magic squares, 10th-order and above

• 1955, in Scripta Mathematica, Walter W. Horner, USA, concluded his paper on the order 8 by mentioning that his "method can be used to construct an addition-multiplication magic square of order 16". But unfortunately he did not publish such a square and it seems unclear how his method can construct one.
• 1992, in the Journal of Combinatorial Mathematics and Combinatorial Computing, the Chinese team Liang Peiji, Sun Rongguo, Ku Tunghsin and Zhu Lie published a method for constructing add-mult magic squares of order mn, where m and n are positive integers different from 1, 2, 3, and 6. This means that their method should be able to construct add-mult magic squares of orders 16, 20, 25, 28, 32, 35,... The only published example was a square having these characteristics:
• Order = 20, S = 42210, P = 3.69e+60, MaxNb = 8020
• 1996, in the Journal of Statistical Planning and Inference, another Chinese team Zhu Jiacheng, Sun Rongguo and Cheng Murong published an add-mult magic square with a method for its construction.
• Order = 18, S = 114400, P = 1.59e+63, MaxNb = 24208
• 1997, in the Journal of Central China Normal University, still another Chinese team Yu Fuxi, Sun Rongguo and Zhang Guiming published an add-mult magic square of order 24 with a method for its construction. Thanks to Zhu Lie, School of Mathematical Sciences, Suzhou University, who informed me of this paper! Zhu Lie is one of the authors of the above paper of 1992.
• May 2006, Su Maoting, China, constructed a better square than the above Peiji-Rongguo-Tunghsin-Lie square, "better" because it uses smaller S, P and MaxNb:
• Order = 20, S = 9460, P = 1.77e+49, MaxNb = 1944
• January 2009, I constructed add-mult squares of various orders, improving the above orders 16, 18, 25 (smaller S, P, MaxNb), and 20 (partially, only smaller MaxNb), but also constructing squares of new orders 10, 12, 14, 15, 21. Many thanks to Eric W. Weisstein, MathWorldJoshua Zucker, Julia Robinson Mathematics Festival, and W. Edwin Clark, Math Dept, University of South Florida, for checking my squares.
• February 2009, I continued with orders 27, 28, 30, 32 and... 1024! The goal of this last square, which has a magic product of 8356 digits (P = 9.94E+8355), is only to prove that very large add-mult squares are possible. All these squares can be downloaded below. Thanks again to Joshua Zucker and W. Edwin Clark for their checking. Good multiprecision math packages were needed in order to check a product of more then 8000 digits!

• April-May-June 2010, Toshihiro Shirakawa found the first known add-mult magic squares of orders 10, 11, 14, 22, 26. On orders 10 and 14, my squares were only semi-magic, their diagonals being mult magic but unfortunately not add magic. Astonishing: my two semi-magic squares of order 14 constructed in January 2009, given in the below Excel file, had already EXACTLY the same characteristics (S, P, MaxNb) = (61740, 7.29e+47, 14976) and (58800, 5.93e+46, 15096) as his two squares, but he is more clever than me... obtaining two add-mult magic diagonals in these squares!
• April-May 2013, Toshihira Shirakawa found the first known add-mult magic squares of prime orders 13, 17, 19. He also found the first known add-mult magic squares of orders 12, 15, 21: my previous squares of these three orders, constructed in 2009, were only semi-magic and had bigger characteristics (S, P, MaxNb). He also found a new best known add-mult magic square of order 14, with better characteristics than his previous square of 2010. Interesting: all his add-mult magic squares of orders ≥ 11 are also the best known multiplicative magic squares!

Here is his best add-mult magic square of order 10:

 165 1599 1890 9520 8424 648 560 4914 3315 6765 6426 9360 216 2255 1365 3549 4675 8856 720 378 2952 1925 4641 6318 240 9840 486 273 5005 6120 4563 162 3280 2520 6545 385 6552 6800 6642 351 2800 8568 6435 117 2214 4590 4797 495 504 7280 1400 5712 7605 243 2706 5610 9963 585 336 3640 9477 198 1640 1680 7735 455 4368 3400 8118 729 1968 2275 9639 7722 120 4920 594 567 5915 4080 7854 4680 144 2665 2835 7371 5525 5904 360 462 195 3321 2310 4760 5616 432 280 6006 6885 7995

And here is an example of my add-mult magic squares:

 1 496 148 130 246 159 357 86 285 540 406 484 276 793 531 400 220 116 432 19 50 413 183 138 286 518 620 15 688 459 689 492 510 645 451 742 481 312 16 558 472 25 115 244 87 264 38 378 854 253 375 590 486 304 528 377 212 205 43 408 434 2 156 111 108 133 132 174 305 92 200 59 992 9 338 444 583 574 430 765 222 78 7 124 51 344 164 265 348 572 171 864 885 250 322 671 225 944 732 299 616 319 810 190 301 102 318 123 104 185 62 8 533 636 816 387 10 930 407 364 69 366 118 175 152 54 145 176 416 333 806 12 473 714 530 615 88 203 162 114 125 236 488 23 228 702 261 704 345 610 826 275 6 186 259 52 41 424 204 215 371 82 258 153 248 5 26 296 549 368 300 767 756 209 660 290 177 150 46 427 232 44 95 216 663 516 656 477 370 390 11 868 435 440 266 594 708 325 207 976 37 208 4 310 306 129 287 106 682 14 260 555 848 369 559 612 270 76 352 29 122 161 75 354 184 61 295 100 57 324 58 308 410 795 561 602 13 744 592 234 172 255 53 328 182 74 372 3 350 649 915 230 396 464 648 247

Table of best known add-mult magic squares

 1 2         3         4Photos 1 and 2 (kindly provided by A. D. Keedwell) : Jószef Dénes (Budapest 1932 - 2002) with Paul Erdös, summer 1990.Photo 3 : A. Donald Keedwell (London 1928 - ) in 1979,Photo 4 : Paul Erdös (Budapest 1913 - Warsaw 1996) in 1992. "Problem 6.3. For what orders n do addition-multiplication magic squares exist?"Jószef Dénes and A. Donald Keedwell, Latin Squares and their Applications (1974) page 489, or (2015) page 346."Many unsolved problems are stated, some classical, some due to the authors, and even some proposed by the writer of this foreword."Paul Erdös, foreword of the above Dénes - Keedwell book (1974) page 5, or (2015) page v.

In this Dénes - Keedwell book, Horner's 8x8 and 9x9 add-mult squares are published on pages 215-216 (or pages 221-222 in the second edition of 2015) and the above quoted problem 6.3 is posed. I do not have the full answer to this Dénes - Keedwell - Erdös problem, but here is a partial answer with a summary of the best known add-mult magic squares that you can download with the Excel file below. When their book was published, only orders 8 and 9 were known. We can note below that orders p (where p is a prime) are difficult to get, but not impossible, as proved by Toshihiro Shirakawa with p = 11, 13, 17, 19.

 Order (*) Square S P MaxNb (**) 3..7 See here 8 Magic (see here) 600 5.14E+13 225 2 9 Magic (see here) 784 2.99E+15 261 2 10 (S) Magic 37800 1.60E+33 9963 1 11 (S) Magic 1034 4.68E+19 238 1 12 (S') Magic 1343 3.23E+22 276 1 13 (S') Magic 1734 1.76E+25 350 2 14 (S') Magic 2158 1.02E+28 406 1 15 (S') Magic 2734 1.14E+31 465 1 16 Magic 5338 1.61E+37 992 1 17 (S') Magic 4101 1.70E+37 627 1 18 Magic 30030 9.97E+52 5848 2 19 (S') Magic 5591 2.02E+43 779 1 20 (M/B) Magic 9460 1.77E+49 1743 2 21 (S') Magic 7393 4.12E+49 987 1 22 (S) Magic 147840 3.98E+78 23214 2 23 Unknown 24 (FRG) Magic 55890 1.50E+77 6402 1 25 Magic 25025 1.76E+70 3025 3 26 (S) Magic 859248 1.11E+106 129297 1 27 Magic 27888 7.53E+75 3753 2 28 Magic 37200 4.98E+81 4321 2 29 Unknown 30 Magic 53968 3.69E+90 7037 1 31 Unknown 32 Magic 60852 4.97E+98 6400 1 1024 (***) Magic 274878169600 9.94E+8355 1072694272 1