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

After additive-multiplicative magic squares of orders 8-9 (also called addition-multiplication magic squares, or even shorter "add-mult"), what about bigger squares?

• 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
• December 2008 and January 2009, I received several add-mult magic squares from Mohamad Moosavi, Islamic Azad University of Rasht, Iran:
• Order = 16, S = 6120, P = 6.31e+37, MaxNb = 1424
• Order = 20, S = 21000, P = 2.50e+55, MaxNb = 4060
• Order = 25, S = 42055, P = 1.32e+75, MaxNb = 6725
• Download add-mult squares of orders 16, 20, 25 by Mohamad Moosavi (Excel file, 35Kb)
   
• 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, MathWorld, Joshua 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:

May 2010: best known add-mult magic square
of order 10, by Toshihiro Shirakawa
S = 37800, P = 1.60e+33, MaxNb = 9963

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:

January 2009: best known add-mult magic square of order 16, by Christian Boyer
S = 5338, P = 1.61e+37, MaxNb = 992

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

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.

Table of best known additive-multiplicative magic squares

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

(*) All these best known squares of orders ≥ 8 by Christian Boyer, Nov. 2005 (orders 8-9) and Jan.-Feb 2009 (orders ≥ 12)
      except (FRG) of order 24 by Yu Fuxi - Sun Rongguo - Zhang Guiming, 1997,
      except (M/B) one out of two of order 20 by Su Maoting, May 2006, the other one being by Christian Boyer, Jan. 2009,
      except (S) of orders 10, 11, 22, 26 by Toshihiro Shirakawa, April-May-June 2010,
      except (S') of orders 12, 13, 14, 15, 17, 19, 21 by Toshihiro Shirakawa, April-May 2013.
(**) The smallest S, P, and MaxNb may occur separately in 1, 2 or 3 different squares of the same order.
(***) This square of order 1024 is only an example proving that very big orders are possible. Other big orders are possible.
(****) If you succeed in getting a smaller P, or a smaller S, or a smaller MaxNb, or a new order, send me a message! Your results will be added in this website.

Download the best known add-mult magic squares of orders ≥ 8 (Excel file, 317Kb)
 
Download the add-mult magic square of order 1024 (Zipped Text file, 5Mb)
   Download the 8356 digits of its product P = 9.94E+8355 (Text file, 100 digits per line, 9Kb)
   Download the factorization of its product P = 2^1023 * 3^1021 * 5^508 * 7^338 * 11^204 * 13^169 * ... * 1038337 (Text file, 10Kb)

Return to the home page http://www.multimagie.com