Some remarks about the definition

Now take a closer look at this pandiagonal magic square of order n=4 with magic sum S=34:

115414
810511
133162
12697

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

115414
810511
133162
12697
115414
810511
133162
12697
115414
810511
133162
12697

This is still true for cycled subsquares:

115414
810511
133162
12697
115414
810511
133162
12697
115414
810511
133162
12697
115414
810511
133162
12697
115414
810511
133162
12697
115414
810511
133162
12697

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.

115414
810511
133162
12697
115414
810511
133162
12697

It even holds for broken diagonals.

Supermagisch-Diagonalen

The following example of a most-perfect magic square of order n=8 will show these properties even more significant.

116173253603744
635047341162722
314193055583942
61524536982524
125282164494833
545938432151831
107262362514635
565740414132029