Most-perfect magic square

A most-perfect magic square of order n is a magic square containing the numbers 1 to n2 with two additional properties: Two 12 × 12 most-perfect magic squares can be obtained adding 1 to each element of: All most-perfect magic squares are panmagic squares.

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 × 1044 essentially different most-perfect magic squares.