Prove that the central cell (the number in the middle cell) of any 3x3 magic square is always one-third the magic constant (the sum of any side, either 2 major diagonals, or either center row in the magic square).
Show that in any larger square (n x n), the central cell does not need to be 1/n the magic constant.
(In reply to re: The second part
From an article at Mathworld the only orders which it is impossible to construct a magic square are 3 and all orders of the form 4k+2. Included in the article is a method which will generate magic squares for any order of the form 6k+/-1.