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.

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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.