Some results on magic squares based on generating magic vectors and R-C similar transformations

Xiaoyang Ma 1 , 2 , Kai-tai Fang 1 , 3 ,  and Yu hui Deng 4
  • 1 Division of Science and Technology, BNU-HKBU United International College, , Zhuhai, China
  • 2 Department of Biostatistics, Georgetown University, , Washington D.C., United States of America
  • 3 The Key Lab of Random Complex Structures and Data Analysis, The Chinese Academy of Sciences, , Beijing, China
  • 4 Division of Science and Technology, BNU-HKBU United International College, , Zhuhai, China

Abstract

In this paper we propose a new method, based on R-C similar transformation method, to study classification for the magic squares of order 5. The R-C similar transformation is defined by exchanging two rows and related two columns of a magic square. Many new results for classification of the magic squares of order 5 are obtained by the R-C similar transformation method. Relationships between basic forms and R-C similar magic squares are discussed. We also propose a so called GMV (generating magic vector) class set method for classification of magic squares of order 5, presenting 42 categories in total.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Candy, A.L. (1939), Construction, Classification and Census of Magic Squares of Order 5 (2nd Edition), Albert L. Candy, Author and Publisher.

  • [2] Chu, K.L and Styan,G. P. H. (2013). Some illustrated comments on Anderson graphs and Greek mythology tables for classic magic squares, presentation at The 22nd International Workshop on Matrices and Statistics (IWMS-2013), Toronto, Canada.

  • [3] Fang, K.T., Y.Y. Luo adn Y.X. Zheng (2015). Classification of magic squares of order 4, Souvenir Booklet of the 24th International Workshop on Matrices and Statistics, 25-28, May 2015, Haikou, China, 84-97, Ed. Jeffrey J. Hunter.

  • [4] Gardner, M. (1975), Mathematical games - a breakthrough in magic squares, and the first perfect magic cube Scientific Americal, Inc., 118-123.

  • [5] Lin, Z.Q., Liu, S.J., Fang, K.T. and Deng, Y.H. (2016), Generation of all magic squares of order 5 and interesting patterns finding, Spec. Matrices 4:110–120.

  • [6] Loly, P., Cameron, I., Trump, W. and Schindel, D. (2009), Magic square spectra, Linear Algebra and its Applications, 430, 2659-2680.

  • [7] Ollerenshaw, D.K. and Bondi H. (1982), Magic squares of order four, Phil. Trans. R. Sac. London, A 306, 443-532.

  • [8] Pickover, Clifford A. (2011), The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, p. 7, ISBN 9781400841516.

  • [9] Schroeppel, R. (1973). The Order 5 Magic Squares Program proposed by Richard Schroeppel, published in Scientific American in January 1976.

  • [10] Wu H. L (2008). Magic Squares and Prime Number: Two Classic Problems in Recreational Mathematics. Beijing, China: Science Press.

  • [11] Xu D. D., Zhang X. B. (2008). Transformation Groups in Order 4Magic Squares, Journal of Nanjing Normal University (Natural Science Edition), Vol.31, No.4, pp.26-28.

  • [12] Xu Z. Y. (1998). Patterns of Number Structures in Order 4Magic Squares, Yan’an Educational Journal, Vol.23, No.2, pp.81-83.

OPEN ACCESS

Journal + Issues

Special Matrices is a peer-reviewed, open access electronic journal that publishes original articles of wide significance and originality in all areas of research involving structured matrices present in various branches of pure and applied mathematics and their noteworthy applications in physics, engineering, and other sciences.

Search