On Systematic Procedures for Constructing Magic Squares

Abstract

In the first part of the paper, a systematic procedure for constructing high-order magic squares as an extension of the lower-order basic magic squares is developed and demonstrated. For a 2N x 2N magic square, one can start with a basic N x N magic square, say, N = 3,4,5 or 7,11,13. Using small 2 x 2 squares, like ( 1,2,3,4) or ( 5,6,7,8), to fill the elements of the basic N x N magic square according to the ordering of the magic square, one develops the 2N x 2N magic square automatically. For a 3N x 3N magic square, one can choose another basic N x N magic square, say, the 4 x 4 magic square. Then, inserting 3 x 3 small squares in the 16 empty spaces according to the ordering of the 4 x 4 magic square, one develops the 12 x 12 magic square. There may be small adjustments in the numbering of ( 1,2,3,4) in the 2 x 2 small square, or of ( 1,2,3,4,5,6,7,8,9) in the 3 x 3 small square to form the proper 2N x 2N or 3N x 3N magic squares. In most cases, the correct results are obtained without adjustments. For N = 4, 4N x 4N produces the 16 x 16 magic square. For N = 5 or 7, using 2 x 2 small squares produces 10 x 10 or 14 x 14 magic squares. Part II of the paper gives a review of other known procedures for constructing odd and even order magic squares with illustrative examples for 4 x 4, 5 x 5, 7 x 7, 10 x 10, 11 x 11, 13 x 13, 15 x 15, 17 x 17, 19 x 19 and 20 x 20 magic squares. One 18 x 18 magic square is given in Part I. In the Appendix, the G and H squares, given by Moschopulus of Constantinople nearly five centuries ago, are discussed concerning symmetrical and pandiagonal squares. It is shown that H can be modified to form H 1 which is both symmetrical and pandiagonal.