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 3-4-56-7, 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.


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.



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.



Pandiagonal multiplicative magic squares, 8th and 9th-order


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