Smallest multiplicative magic squares, 6th and 7th-order

After multiplicative magic squares of orders 3-4-5, and before multiplicative magic squares of orders 8-9 and >=10,  here are my results on orders 6-7.

In 1983, Debra K. Borkovitz and Frank K.-M. Hwang published in Discrete Mathematics this 6x6 multiplicative magic square:

As far as I know, it was the smallest published 6x6 example. The 6x6 case is difficult: the latin squares method, described for 4x4 and 5x5 multiplicative squares, does not work for 6x6. The famous "36-officers problem" of Euler has no solution, as first proved by Gaston Tarry in 1900. Is it possible to construct smaller squares than the B&W example?

The answer is yes: the smallest possible 6x6 multiplicative magic product is P = 25 945 920, more than 70 times smaller than the B&H example. And the smallest possible maximum number with that product is more than 16 times smaller than the B&H example. Here is one of the numerous examples with this smallest P and its best max nb:

2005: 6x6 multiplicative square,
Smallest P = 25 945 920,
Max nb = 78

1	22	39	54	28	20
12	65	18	8	3	77
30	27	14	52	11	4
42	16	6	33	15	13
26	21	44	5	24	9
66	2	10	7	78	36

For 3x3, 4x4, 5x5 and 7x7 squares, the smallest possible maximum number is always possible with the smallest P. But it is not the case for 6x6 squares! Here is an example with the smallest possible maximum number for 6x6 squares, but needing a bigger P:

2005: 6x6 multiplicative square,
P = 39 916 800,
Smallest Max nb = 66

8	45	42	66	1	40
50	56	44	3	54	2
16	33	5	28	30	18
21	10	15	9	64	22
27	4	6	32	55	35
11	12	48	25	7	36

More magic: it is possible to have some additive properties in a 6x6 multiplicative square! Here is an example with the same P:

2005: 6x6 multiplicative square
with 6 additive-multiplicative rows,
P = 39 916 800, S = 173,
Max nb = 88

1	36	33	48	35	20
54	55	2	10	24	28
50	14	72	22	12	3
88	16	30	7	5	27
8	6	70	60	18	11
21	15	4	9	44	80

All rows of this multiplicative magic square are additive-multiplicative magic:

• When you multiply the 6 integers of each row, you get the same product P
• AND when you add the same 6 integers of each row, you get the same sum S = 173

6x6 pandiagonal additive magic squares (using consecutive integers) are impossible. But 6x6 pandiagonal multiplicative magic square are possible! In 1913, Harry A. Sayles published this 6x6 pandiagonal multiplicative magic square, meaning that all the broken diagonals have the same product P.

1913: 6x6 pandiagonal
multiplicative square,
P = 101 559 956 668 416,
Max nb = 46 656, by Sayles[292, 560]

729	192	9	46656	3	576
32	486	2592	2	7776	162
11664	12	144	2916	48	36
1	15552	81	64	243	5184
23328	6	288	1458	96	18
16	972	1296	4	3888	324

It is also a most-perfect magic square: all its 2x2 subsquares (as for example the green one) have the same product P'. And another supplemental feature: all its 3x3 subsquares (as for example the blue one) have the same product P''.

Is it possible to do better? Yes. Here is my best 6x6 pandiagonal multiplicative magic square with the smallest known product P, more than 30 times smaller than P of the Sayles's example. And it has also the same 2x2 and 3x3 subsquares features.

2006: 6x6 pandiagonal multiplicative square,
P = 2 985 984 000 000,
Max nb = 4 800

5	720	160	45	80	1440
4800	12	150	192	300	6
9	400	288	25	144	800
320	180	10	2880	20	90
75	48	2400	3	1200	96
576	100	18	1600	36	50

And here are my best 6x6 pandiagonal multiplicative magic square with the smallest known Max nb, more than 10 times smaller than the Max nb of the Sayles's example. And it has also again the same 2x2 and 3x3 subsquares features.

2006: 6x6 pandiagonal multiplicative square,
P = 85 766 121 000 000,
Max nb = 4 410

14	2100	63	350	84	1575
4410	15	980	90	735	20
50	588	225	98	300	441
126	525	28	3150	21	700
490	60	2205	10	2940	45
450	147	100	882	75	196

In December 2007, Lee Morgenstern proved that the magic product of any 6x6 pandiagonal multiplicative magic square is always a 6th power. Look at the 3 above squares, their products are:

• P = 101 559 956 668 416 = 216⁶
• P = 2 985 984 000 000 = 120⁶
• P = 85 766 121 000 000 = 210⁶

See here his proof of the 6th power.

The ten smallest possible products for 6x6 multiplicative magic squares are:

The ten smallest magic products P
of 6x6 multiplicative magic squares

#	P	= 2^	· 3^	· 5^	· 7^	· 11^	· 13^	· 17^	· 19^
1	25 945 920	6	4	1	1	1	1	0	0
2	26 611 200	9	3	2	1	1	0	0	0
3	28 828 800	7	2	2	1	1	1	0	0
4	29 937 600	6	5	2	1	1	0	0	0
5	31 449 600	9	3	2	1	0	1	0	0
6	33 264 000	7	3	3	1	1	0	0	0
7	33 929 280	6	4	1	1	1	0	1	0
8	34 594 560	8	3	1	1	1	1	0	0
9	34 927 200	5	4	2	2	1	0	0	0
10	35 380 800	6	5	2	1	0	1	0	0

For more terms: see the 6x6 list referenced in Jan. 2006 under the number A113026 in the Neil Sloane's Encyclopedia of Integer Sequences, AT&T Research.

Smallest 7th-order multiplicative magic squares

What about 7x7 multiplicative squares? It seems that the problem is new, I have not seen any published 7x7 example.

Here is an example using the smallest possible product, and using the smallest possible maximum number:

2005: 7x7 multiplicative square,
Smallest P = 3 632 428 800,
Smallest Max nb = 91

11	27	1	64	60	49	65
52	35	36	55	2	12	42
28	48	39	10	70	33	3
25	44	56	9	18	26	14
6	5	77	78	84	8	30
63	4	40	21	13	20	66
24	91	15	7	22	45	16

As for 6x6 squares, it is possible to have some additive properties in a 7x7 multiplicative magic square. But this time directly with the same smallest P, and same smallest maximum number!

2005: 7x7 multiplicative square
with 7 additive-multiplicative rows,
Smallest P = 3 632 428 800, S = 238,
Smallest Max nb = 91

35	48	1	39	40	33	42
77	65	20	16	36	3	21
4	56	54	7	26	25	66
9	14	44	24	6	91	50
52	5	70	63	22	18	8
12	11	84	10	15	28	78
60	27	13	55	49	32	2

I do not know if it is possible to have 6x6 or 7x7 full (rows+columns+diagonals) additive-multiplicative magic squares, as it is possible for 8x8 and 9x9 additive-multiplicative squares.

But it is possible to have 7x7 pandiagonal multiplicative magic squares, meaning that all the broken diagonals are also multiplicative magic. Here are two examples, probably using the smallest possible P (first example) and the smallest possible max nb (second example) of 7x7 pandiagonal squares:

2005: 7x7 pandiag. multiplicative square,
P = 8 821 612 800, Max nb = 136

22	36	6	13	30	35	136
21	85	88	18	4	78	10
52	60	7	51	55	72	2
45	8	26	40	42	17	33
102	11	27	5	104	20	28
80	14	68	66	9	3	65
1	39	50	56	34	44	54

 

2005: 7x7 pandiag. multiplicative square,
P = 17 643 225 600,Max nb = 119

54	17	39	50	7	32	44
112	22	36	102	13	30	5
10	3	80	77	18	68	78
34	52	60	1	48	55	63
33	45	119	26	40	6	16
4	96	11	27	85	91	20
65	70	2	64	66	9	51

The ten smallest possible products for 7x7 multiplicative magic squares are:

The ten smallest magic products P
of 7x7 multiplicative magic squares

#	P	= 2^	· 3^	· 5^	· 7^	· 11^	· 13^	· 17^	· 19^
1	3 632 428 800	8	4	2	2	1	1	0	0
2	4 151 347 200	11	4	2	1	1	1	0	0
3	4 410 806 400	7	4	2	1	1	1	1	0
4	4 540 536 000	6	4	3	2	1	1	0	0
5	4 670 265 600	8	6	2	1	1	1	0	0
6	4 750 099 200	8	4	2	2	1	0	1	0
7	4 843 238 400	10	3	2	2	1	1	0	0
8	4 929 724 800	7	4	2	1	1	1	0	1
9	5 145 940 800	6	3	2	2	1	1	1	0
10	5 189 184 000	9	4	3	1	1	1	0	0

For more terms: see the 7x7 list referenced in Jan. 2006 under the number A113027 in the Neil Sloane's Encyclopedia of Integer Sequences, AT&T Research.

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