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.

There are 31 positive integers less than 32.

7 is the largest prime that can be a factor in the matrix, because it appears 4 times as a factor of numbers under 32. (7, 14, 21, 28).

So we can eliminate any number with a prime factor over 7

(11, 13, 17, 19, 22, 23, 26, 29, 31).

Also, we can eliminate 25 (5^2) because 5 is only a factor 5 other times: 5, 10, 15, 20, 30. (Of these, 0 or 4 are in the grid).

This leaves us with just 21 candidate numbers:

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

Their product (if I have done it correctly is 2^24 * 3^13 * 5^5 * 7^4. The maximum column product is 2^6 * 3^3 * 5 * 7

*Edited on ***February 17, 2016, 5:16 pm**