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On systematic procedures for constructing magic squares. (English) Zbl 0596.05020

Summary: 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\(\times 2N\) magic square, one can start with a basic \(N\times N\) magic square, say, \(N=3,4,5\) or 7,11,13. Using small \(2\times 2\) squares, like (1,2,3,4) or (5,6,7,8), to fill the elements of the basic \(N\times N\) magic square according to the ordering of the magic square, one develops the 2N\(\times 2N\) magic square automatically. For a 3N\(\times 3N\) magic square, one can choose another basic \(N\times N\) magic square, say, the \(4\times 4\) magic square. Then, inserting \(3\times 3\) small squares in the 16 empty spaces according to the ordering of the \(4\times 4\) magic square, one develops the 12\(\times 12\) magic square. There may be small adjustments in the numbering of (1,2,3,4) in the \(2\times 2\) small square, or of (1,2,3,4,5,6,7,8,9) in the \(3\times 3\) small square to form the proper 2N\(\times 2N\) or 3N\(\times 3N\) magic squares. In most cases, the correct results are obtained without adjustments. For \(N=4\), 4N\(\times 4N\) produces the 16\(\times 16\) magic square. For \(N=5\) or 7, using \(2\times 2\) small squares produces 10\(\times 10\) or 14\(\times 14\) 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 \(4\times 4\), \(5\times 5\), \(7\times 7\), 10\(\times 10\), 11\(\times 11\), 13\(\times 13\), 15\(\times 15\), 17\(\times 17\), 19\(\times 19\) and 20\(\times 20\) magic squares. One 18\(\times 18\) 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.

MSC:

05B99 Designs and configurations
05B15 Orthogonal arrays, Latin squares, Room squares

Keywords:

magic squares
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References:

[1] Andrews, W. S., Magic Squares and Cubes (1908), Open Court: Open Court Chicago, and an introduction by Carus P.,Frieson L.S., Browne C.A. Jr., Carus P.
[2] Ball, W. W., Mathematical Recreations and Essays (1914), Macmillan: Macmillan London · JFM 45.0371.03
[3] Cammaan, S., The evolution of magic squares in China, J. Am. Oriental Soc., Vol. 80 (1960), Yale University: Yale University New Haven, No. 2
[4] Cammaan, S., Islamic and Indian Magic Squares (1969), University of Chicago Press: University of Chicago Press Chicago
[5] Candy, A. L., Construction, Classification and Census of Magic Squares of Order Five (1940), University of Nebraska: University of Nebraska Lincoln · JFM 64.0989.06
[6] Candy, A. L., Pandiagonal Magic Squares of Prime Order (1940), University of Nebraska: University of Nebraska Lincoln · Zbl 0061.06506
[7] Candy, A. L., Pandiagonal Magic Squares of Composite Order (1941), University of Nebraska: University of Nebraska Lincoln · Zbl 0060.08711
[8] Candy, A. L., Supplement to Pandiagonal Magic Squares of Prime Order (1942), University of Nebraska: University of Nebraska Lincoln · Zbl 0060.08712
[9] Carcopino, J., Etudes d’histoire chrétienne (1963-65), A. Michele: A. Michele Paris
[10] McClintock, E., On the most perfect forms of magic squares with methods for their production, Am. J. Mathematics (1897), Baltimore · JFM 28.0198.03
[11] de Haas, K. H., Frennicle’s 880 Basic Magic Squares of 4 x 4 Cells, Normalized, Indexed and Inventoried (1935), D. van Sijn & Zonen: D. van Sijn & Zonen Rotterdam
[12] Kraitchik, M., Traite des carrés magiques (1930), Gauthier-Villars: Gauthier-Villars Paris
[13] Liharzik, F., Das Quadrat, die Grundlage aller Proportionalität in der Natur und das Quadrat aus der Zahl sieben, die Uridee des Menschlichen Körperbaues (1865), Herzfeld & Bauer: Herzfeld & Bauer Wien
[14] Needham, J., Science and Civilisation in China, Mathematics (1959), Cambridge University Press: Cambridge University Press Cambridge, Vol. 3 · Zbl 0099.24101
[15] Schubert, H. C.H., Mathematical Essays and Recreation (1898), Open Court: Open Court Chicago, 1910,1917
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