Recently, I saw a question in see here which is so interesting for me. This question is as follows:

Is it possible to fill the $121$ entries in an $11×11$ square with the values $0,+1,−1$, so that the row sums and column sums are $22$ distinct numbers?

This problem is interesting for me because its definition is reverse of the magic square. I think for solve this problem we must use something like design theory. Also, for $n=6$ this problem have an answer. My next question is:

For which integer number $n$, this problem has a solution and is this problem well known in math?

I will be so thankful for any helpful comments and answers.