Symmetrical magic squares

An important, relatively frequent special form of magic squares are symmetrical squares, where all pairs of cells diametrically equidistant from the center have the same sum n² + 1. These number pairs are said to be complementary.

Symmetrical magic squares have been intensively investigated. Some important results are given here:

• The only magic square of the third order is symmetrical.
• The center cell of an odd symmetrical square is always equal to the middle number of the series.
• There are 48 symmetrical squares of fourth order.
• The minimum order for a magic square that is both symmetrical and pandiagonal, is n=5.
• There is no symmetrical square of single-even order.
• Each symmetrical magic square is also semi-pandiagonal. But the converse is not true: not every semi-pandiagonal square is also symmetrical.