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 3-4-5, 6-7, 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.

Read the biography of Walter W. Horner
 

1955: 8x8 additive-multiplicative magic square,
by Walter W. Horner,
P = 2 058 068 231 856 000, S = 840,
Max nb = 261

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

1952: 9x9 additive-multiplicative magic square,
by Walter W. Horner,
P = 5 804 807 833 440 000, S = 848,
Max nb = 290

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:
   

2005: 8x8 additive-multiplicative magic square,
P = 51 407 948 592 000, S = 760,
Max nb = 333

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

2005: 8x8 additive-multiplicative magic square,
P = 67 463 283 888 000, S = 600,
Max nb = 225

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:
   

2005: 9x9 additive-multiplicative magic square,
 P = 2 987 659 715 040 000, S = 784,
Max nb = 310

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

2005: 9x9 additive-multiplicative magic square,
P = 7 349 391 483 033 600, S = 840,
Max nb = 261

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:
   

2005: 8x8 multiplicative magic square
Smallest P = 670 442 572 800, Max nb = 228
by Luke Pebody

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

2005: 9x9 multiplicative magic square
Smallest P = 140 792 940 288 000, Max nb = 323
by Luke Pebody 

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.
   

2005: 8x8 multiplicative magic square,
P = 4 693 098 009 600, Smallest Max nb = 160

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




2005: 9x9 multiplicative magic square,
P = 3 238 237 626 624 000, Smallest Max nb = 225

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).
   

2013: 8x8 multiplicative magic square
P = 1 307 674 368 000
Smallest known Max nb = 117
by Toshihiro Shirakawa

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




2013: 9x9 multiplicative magic square
P = 266 765 571 072 000
Smallest known Max nb = 153
by Toshihiro Shirakawa

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.
   

2006: 8x8 pandiagonal multiplicative magic square,
P = 3 266 533 992 960 000, Max nb = 7 560.

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




2006: 9x9 pandiagonal multiplicative magic square,
P = 794 280 046 581 000 000 000, Max nb = 44 100.

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.
   

2007: 8x8 pandiagonal multiplicative magic square,
Smallest known P = 42 074 422 790 400,
Max nb = 546,
by J. Wroblewski and C. Boyer

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




2007: 8x8 pandiagonal multiplicative magic square,
P = 398 337 730 560 000,
Smallest known Max nb = 528,
by J. Wroblewski and C. Boyer

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




2007: 9x9 pandiagonal multiplicative magic square,
Smallest known P = 294 451 250 429 952 000,
Max nb = 1 760,
by J. Wroblewski and C. Boyer

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:

2017: 9x9 pandiagonal multiplicative magic square,
Smallest known P = 294 451 250 429 952 000,
Max nb = 1 760,
by Elbert Krison, improving Wroblewski-Boyer's square

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.

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