The Special Table

Question

Consider a $n\times n$ table which is filled with numbers $1,2,3,...,n^2$ (in no particular order).We'd call such table "Special" if the sum of each column and each row and each diameter is a constant number.For what $n$s do we have a Special Table?

Answer 1

What you call a "special table" is commonly known as a (normal) "magic square".

The Wikipedia article on magic squares links to various methods for constructing them explicitly for all side lenghts except $2$. (One easily sees that the $n=2$ case is impossible).