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Semi-magic squares and their orthogonal complements

Published online by Cambridge University Press:  01 August 2016


Fernando Mayoral
Affiliation:
Depart. de Matemática Aplicada II, E.S, Ingenieros Industrials, Av. Reina Mercedes sin, 41012-Sevilla, Spain

Extract

Not long ago I came across the following quiz in the Sunday supplement of the Spanish newspaper El País: Check that it does not matter how you take any four numbers from the following array; provided that no two of them lie in the same row or column, their sum is always the same.

This reminded me of magic squares: square matrices where the sum of the elements in each line (row or column) and diagonal is always the same. Magic squares are a topic in recreational mathematics. Usually one looks for magic squares with elements that are integer, positive and different from each other, e.g. 1,2,…, n 2 for a n × n as in

According to a Chinese legend, matrix A appeared engraved on the carapace of a turtle about 2000 BC. On the other hand, matrix B appears in Albrecht Dürer’s engraving Melencholia (1514) [1].


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Articles
Copyright
Copyright © The Mathematical Association 1996

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References

1. Andrews, W.S., Magic squares and cubes, Dover, New York (1960).Google Scholar
2. Halmos, P.R., Finite-dimensional Vector Spaces, van Nostrand, Princeton (1958). New edition: Springer-Verlag, Berlin Heidelberg and New York (1974).Google Scholar
3. Strang, G., Introduction to Linear Algebra, Wellesley Cambridge Press, Wellesley MA (1983).Google Scholar
4. Weiner, L.M., The algebra of semi-magic squares, Amer. Math. Monthly 62 (1955) pp. 237239.CrossRefGoogle Scholar
5. den Essen, Arno van, Magic Squares and Linear Algebra, Amer. Math. Monthly 97 (1990) pp. 6062.CrossRefGoogle Scholar

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