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Dot Products and Matrix Properties of 4×4 Strongly Magic Squares

Received: 4 November 2016     Accepted: 27 December 2016     Published: 13 February 2017
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Abstract

Magic squares have been known in India from very early times. The renowned mathematician Ramanujan had immense contributions in the field of Magic Squares. A magic square is a square array of numbers where the rows, columns, diagonals and co-diagonals add up to the same number. The paper discuss about a well-known class of magic squares; the strongly magic square. The strongly magic square is a magic square with a stronger property that the sum of the entries of the sub-squares taken without any gaps between the rows or columns is also the magic constant. In this paper a generic definition for Strongly Magic Squares is given. The matrix properties of 4×4 strongly magic squares dot products and different properties of eigen values and eigen vectors are discussed in detail.

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 2)
DOI 10.11648/j.ijtam.20170302.13
Page(s) 64-69
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Strongly Magic Square (SMS), Dot Products of SMS, Eigen Values of SMS, Rank and Determinant of SMS

References
[1] Schuyler Cammann, Old Chinese magic squares. Sinologica 7 (1962), 14–53.
[2] Andrews, W. S. Magic Squares and Cubes, 2nd rev. ed. New York: Dover, 1960.
[3] Claudia Zaslavsky, Africa Counts: Number and Pattern in African Culture. Prindle, Weber & Schmidt, Boston, 1973.
[4] Paul C. Pasles. Benjamin Franklin’s numbers: an unsung mathematical odyssey. Princeton UniversityPress, Princeton, N. J., 2008.
[5] C. Pickover. The Zen of Magic Squares, Circles and Stars. Princeton University Press, Princeton, NJ, 2002.
[6] Bruce C. Berndt, Ramanujan’s Notebooks Part I, Chapter 1 (pp 16-24), Springer, 1985.
[7] T. V. Padmakumar “Strongly Magic Square”, Applications Of Fibonacci Numbers Volume 6 Proceedings of The Sixth International Research Conference on Fibonacci Numbers and Their Applications, April 1995.
[8] Charles Small, “Magic Squares Over Fields” The American Mathematical Monthly Vol. 95, No. 7 (Aug. - Sep., 1988), pp. 621-625.
[9] Neeradha. C. K, Dr. V. Madhukar Mallayya “Generalized Form Of A 4x4 Strongly Magic Square” IJMMS, Vol. 12, No. 1 (January-June; 2016), pp 79-84.
[10] A. Mudgal, Counting Magic Squares, Undergraduate thesis, IIT Bombay, 2002.
Cite This Article
  • APA Style

    Neeradha. C. K., V. Madhukar Mallayya. (2017). Dot Products and Matrix Properties of 4×4 Strongly Magic Squares. International Journal of Theoretical and Applied Mathematics, 3(2), 64-69. https://doi.org/10.11648/j.ijtam.20170302.13

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    ACS Style

    Neeradha. C. K.; V. Madhukar Mallayya. Dot Products and Matrix Properties of 4×4 Strongly Magic Squares. Int. J. Theor. Appl. Math. 2017, 3(2), 64-69. doi: 10.11648/j.ijtam.20170302.13

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    AMA Style

    Neeradha. C. K., V. Madhukar Mallayya. Dot Products and Matrix Properties of 4×4 Strongly Magic Squares. Int J Theor Appl Math. 2017;3(2):64-69. doi: 10.11648/j.ijtam.20170302.13

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  • @article{10.11648/j.ijtam.20170302.13,
      author = {Neeradha. C. K. and V. Madhukar Mallayya},
      title = {Dot Products and Matrix Properties of 4×4 Strongly Magic Squares},
      journal = {International Journal of Theoretical and Applied Mathematics},
      volume = {3},
      number = {2},
      pages = {64-69},
      doi = {10.11648/j.ijtam.20170302.13},
      url = {https://doi.org/10.11648/j.ijtam.20170302.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170302.13},
      abstract = {Magic squares have been known in India from very early times. The renowned mathematician Ramanujan had immense contributions in the field of Magic Squares. A magic square is a square array of numbers where the rows, columns, diagonals and co-diagonals add up to the same number. The paper discuss about a well-known class of magic squares; the strongly magic square. The strongly magic square is a magic square with a stronger property that the sum of the entries of the sub-squares taken without any gaps between the rows or columns is also the magic constant. In this paper a generic definition for Strongly Magic Squares is given. The matrix properties of 4×4 strongly magic squares dot products and different properties of eigen values and eigen vectors are discussed in detail.},
     year = {2017}
    }
    

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Author Information
  • Dept. of Science & Humanities, Mar Baselios College of Engineering & Technology, Thiruvananthapuram, Kerala, India

  • Department of Mathematics, Mohandas College of Engineering & Technology, Thiruvananthapuram, Kerala, India

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