Conditions for magic square.

Question

So I've messing around with magic squares and something occured to me: Assume we have a nxn grid of numbers which respects the sum conditions of a magic square as in it has the appropriate column, diagonal and row sums. Is it guaranteed that it is a magic square or do I also need to check whether it contains the appropriate numbers? Basically does a non-magic square grid that respects the summing conditions of a magic square grid exist? If such a grid exist, does it also exist if all the numbers in the grid are guaranteed to be positive? I was thinking about this and I haven't figured out a way to prove any of these statements wrong or true. Any help is much appreciated!

Answer 1

If I understand what you are asking, you want to know if any arrangement of numbers in a square, where the rows, columns, and diagonals all sum to the same amount, that is not a magic square. If this is the case, then the only other requirement is that the numbers in the square all be distinct. However, there is a simple counterexample: $$ \begin{array}{ccc} 1&1&1\\1&1&1\\1&1&1 \end{array} $$ Here, every row, column, and diagonal sums to 3, but it is not a magic square because not every number is distinct (in fact, none of them are!).