Apart from the trivial case of the first order square, most-perfect magic squares are all of order 4n. In their book, Kathleen Ollerenshaw and David S. Brée give a method of construction and enumeration of all most-perfect magic squares. They also show that there is a one-to-one correspondence between reversible squares and most-perfect magic squares.
For n = 36, there are about 2.7 × 1044essentially different most-perfect magic squares.
References
Kathleen Ollerenshaw, David S. Brée: Most-perfect Pandiagonal Magic Squares: Their Construction and Enumeration, Southend-on-Sea : Institute of Mathematics and its Applications, 1998, 186 pages, ISBN 0-905091-06-X
T.V.Padmakumar, Number Theory and Magic Squares, Sura booksArchived 2010-02-25 at the Wayback Machine, India, 2008, 128 pages, ISBN 978-81-8449-321-4