Smallest multiplicative magic squares, 10th-order and above


After multiplicative magic squares of orders 3-4-56-78-9, here are my results on orders from 10 to 17.

My best 10x10 examples, "best" meaning with smallest product and/or smallest maximum number, are:


It should be possible to construct better 10x10 squares. Luke Pebody, England, also studied 10x10 multiplicative magic squares: he thinks that the smallest possible P is 43,716,207,959,424,000 (more than 6 times smaller than my best P above), but has not yet succeeded to construct such a square.

Pandiagonal additive magic squares (using consecutive integers) of order 10 are impossible, and more generally all 4k+2 orders. But pandiagonal multiplicative magic squares of order 10 are possible. This example is also a most-perfect square, because all its 2x2 subsquares (as for example the blue one) have the same product P'. And all its 5x5 subsquares (as for example the green one) have the same product P''.

With Jaroslaw Wroblewski, Poland, I constructed in May-June 2007 two better pandiagonal squares of order 10 having smaller P and Max nb than the previous one. However, they are no longer most-perfect squares. Here they are:

I also constructed 11x11 pandiagonal multiplicative magic squares. I think that it will not be possible to construct better 11x11 pandiagonal squares.


I also constructed examples from 12x12 to 17x17. It would be boring to directly display all of them in this page, even if I am proud for example of the 15x15 pandiagonal square, for me the most difficult to construct. But you can get them, and all the other squares of this page:

The pandiagonal results are summarized in this table. One of the two biggest squares of these Excel files:

In the files, you will find another 17x17 pandiagonal multiplicative magic square, with a bigger P, but with a smaller max nb = 1003.

Many thanks to Edwin Clark, Don Reble, and GŁnter Stertenbrink for checking these big multiplicative squares, confirming that they have all the announced properties... and that all their integers are really distinct!


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