Some remarks about the definition
Now take a closer look at this pandiagonal magic square of order n=4 with magic sum S=34:
Any four adjacent integers forming a 2x2-subsquare sum to S=34, for example
1 + 15 + 8 + 10=34 oder 16 + 2 + 9 + 7=34
This is still true for cycled subsquares:
Second, any pair of integers distant n/2=2 along a diagonal sum to
T=n2 + 1
This means that these integers are complementary numbers. It doesn't matter what kind of diagonal you look at. For all most-perfect magic squares of order n=4 you will get the sum T=17.
It even holds for broken diagonals.
The following example of a most-perfect magic square of order n=8 will show these properties even more significant.