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

I do not know if it is possible to have 6x6 full (rows+columns+diagonals) additive-multiplicative magic squares, look here.

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 = 2166
• P = 2 985 984 000 000 = 1206
• P = 85 766 121 000 000 = 2106
• P = 46 656 000 000 = 606

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 606. 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

On 7x7 full (rows+columns+diagonals) additive-multiplicative magic squares, look here.

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.