**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:

1 |
140 |
60 |
21 |
180 |
63 |

315 |
35 |
12 |
84 |
45 |
4 |

126 |
18 |
210 |
30 |
70 |
2 |

10 |
90 |
42 |
6 |
14 |
360 |

252 |
28 |
105 |
15 |
36 |
5 |

20 |
9 |
3 |
420 |
7 |
1260 |

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&H 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:

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:

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:

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, which means that all the broken diagonals have
the same product P.

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 my smallest product P, more than 30 times smaller than P of Sayles's example. And it has also the same 2x2 and 3x3 subsquares features.

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 my smallest Max nb, more than 10 times smaller than the Max nb of Sayles's example. And it also has again the same 2x2 and 3x3 subsquares features.

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 May 2012, **Radko Nachev **beat my
above results with this very nice square. His square is no longer most-perfect, but is still
3x3 magic. Radko Nachev, born in 1950, Sofia, Bulgaria, is an assistant civil engineer working at the Department
of Transportation of New York City, USA.

1 |
36 |
600 |
80 |
120 |
225 |

300 |
1800 |
3 |
5 |
144 |
40 |

200 |
4 |
360 |
150 |
15 |
72 |

45 |
30 |
16 |
3600 |
100 |
6 |

720 |
25 |
90 |
12 |
2 |
1200 |

24 |
240 |
50 |
18 |
900 |
10 |

But in February 2014, **Oscar Lanzi** cleverly improved Nachev's
square! Simply replacing 3^n factors by 3^(2-n), he obtained this square with
the same P, but with a better MaxNb of 1800 instead of 3600. Oscar Lanzi is an
engineer at ArcelorMittal LLC, East Chicago (Indiana, USA), Global R&D.
Born 1960 in Canton (Ohio), currently living in Chicago.

9 |
4 |
600 |
720 |
120 |
25 |

300 |
200 |
3 |
45 |
16 |
360 |

1800 |
36 |
40 |
150 |
15 |
8 |

5 |
30 |
144 |
400 |
900 |
6 |

80 |
225 |
10 |
12 |
18 |
1200 |

24 |
240 |
450 |
2 |
100 |
90 |

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 squares above, their products are:

- P = 101 559 956 668 416 = 216
^{6} - P = 2 985 984 000 000 = 120
^{6} - P = 85 766 121 000 000 = 210
^{6} - P = 46 656 000 000 = 60
^{6}

See here his proof of the 6th power.

In May 2015, **Oscar Lanzi** proposed this proof: the smallest possible magic
product of 6x6 pandiagonal multiplicative magic squares is 60^{6}. See here his proof (PDF
file of 655Kb).

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

# |
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 On-Line Encyclopedia of Integer Sequences, OEIS Foundation.

**Smallest 7th-order multiplicative magic squares**

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

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

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!

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:

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 |

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:

# |
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 On-Line Encyclopedia of Integer Sequences, OEIS Foundation.

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