### Some remarks about the definition

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

 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7

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

 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7
 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7
 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7

This is still true for cycled subsquares:

 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7
 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7
 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7
 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7
 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7
 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7

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.

 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7
 1 15 4 14 8 10 5 11 13 3 16 2 12 6 9 7

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.

 1 16 17 32 53 60 37 44 63 50 47 34 11 6 27 22 3 14 19 30 55 58 39 42 61 52 45 36 9 8 25 24 12 5 28 21 64 49 48 33 54 59 38 43 2 15 18 31 10 7 26 23 62 51 46 35 56 57 40 41 4 13 20 29