I was shown a 4x4 matrix whose entries are distinct positive integers below
32, whose row sums are the same (say S), and whose column products are the same (P).

I believe that there are numerous solutions fitting the above definition, and challenge the solvers to find as many as possible - the values of S and P are up to you.

Rem : It is possible to find a solution or two without a computer, but listing** all** possible solutions is IMHO a hard nut even for a software-based approach.

I have previously gotten us to just 21 candidate numbers:

1 - 10, 12, 14-16, 18, 20, 21, 24, 27, 28, 30.

These have a product of 2^24 * 3^13 * 5^5 * 7^4.

If you remove the multiples of 7 (i.e, 7, 14, 21, 28)

then the remaining numbers have a product of 2^21 * 3^12 * 5^5.

Clearly, the 5th number that must go is 2*5

The remaining numbers have a sum of 176, which is a multiple of 4.

So it may be possible to arrange

1 - 6, 8, 9, 12, 15, 16, 18, 20, 24, 27, 30

so that they have a row sum of 44 and a column product of 2^5*3^4*5.

I have to run, but it should not be hard to work out, as 20, 24, 27 and 30 all need to be in separate rows and columns. And then 18 needs to be in the same row as 20. And then 16 needs to be in the same row as 24. No, too many high numbers. 15 cannot be in the same row as 27.

So this does not work.

So the 4 multiples of 7 must be in the matrix.

*Edited on ***February 17, 2016, 5:55 pm**