Magic Squares and Latin Squares

Year 2013, Volume: 10 Issue: 1, 17 - 28, 15.07.2013

Fikri Akdeniz

Abstract

In this paper, magic squares which have been known for 4000 years are presented. Especially, Moore-Penrose inverse and eigenvalue properties of 2n x 2n (n≥2) singular magic squares are investigated. Furthermore, Latin square design models are illustrated visually by special stamps.

Keywords

Latin Square, Moore-Penrose İnverse, Magic Constant, Magic Square, Semi-Magic Square

Sihirli Kareler ve Latin Kareler

Abstract

Bu makalede 4000 yıldır bilinen sihirli kareler ele alınmıştır. Özellikle singular matrisler veren 2nx2n (n≥2) boyutlu sihirli matrislerin Moore-Penrose inversi ve özdeğerlerinin özellikleri incelenmiştir. Ayrıca, Latin kare tasarım modelleri pullar yardımıyla görsel olarak verilmiştir.

Keywords

Latin Kare, Moore-Penrose İnvers, Sihirli Sabit, Sihirli Kare

References

• Akdeniz, F., 2011. Sihirli karelerin gizeminden Nye Gezinti İstatistik Dergisi, Yıl:1, Sayı: 3, 12-17.
• Ataöv, T., 2002. Hintli Bir Matematik Dahisi. Cumhuriyet Bilim Teknik, Sayı: 773.
• Booth, A. D., Booth, K. H. V. 1955. On Magic Squares. The Mathematical Gazette, 39, 132-133.
• Henrich, C. J., 1991. Magic squares and linear algebra. Amer. Math. Monthly 98, 481-488.
• Ka Lok Chu, Puntanen, S., Styan, G. P. H., 2009. Some comments on philatelic Latin squares from Pakistan, Pak. J. Statist. 25(4), 427-471.
• Ka Lok Chu, Drury, S. W., Styan, G. P. H., Trenkler, G., 2011. Magic Moore- Penrose inverses and philatelic magic squares with special emphasis on the Daniels-Zlobec magic square, Croatian Operational Research Review 2, 4-13.
• Loly, P. D., Styan, G. P. H., 2010. Comments on 5x5 Philatelic Latin SquaresCHANCE Vol.23(2), 1-10.
• Pasles, P. C., 2001. The lost squares of Dr. Franklin: Ben Franklin’s missing squares and the secret of the magic circle. The American Mathematical Monthly, 108, 489-511.
• Rao, C. R., Mitra, S. K., 1971. Generalized Inverse of Matrices and its Applications, New York: John Wiley.
• Schmidt, K. and Trenkler, G., 2001. The Moore-Penrose inverse of a semi- magic square is semi-magic. International Journal of Mathematical Education in Science and Technology, 32, 624-629.
• Styan, G. H. P. Boyer, C., Ka Lok C., 2009. Some comments on Latin squares and on Graeco- Latin squares, illustrated with postage stamps and old playing cards. Stat. Papers 50: 917-941.
• Trenkler, G., 1994. Singular magic squres. International Journal of Mathematical Education in Science and Technology, 25, 595-597.
• William L. S., 1978. Mathematics and Science: An Adventure in Postage Stamps National Council of Teachers of Mathematics, Reston, Virginia, 2.