Enhanced Technique of Constructing Multiple ODD Magic Square Matrices

Salam Samarendra Singh *

Department of Mathematics, G.P. Women’s College, DMU, Imphal- 795001, India.

Ngasham Amarjit Singh

Department of Mathematics, DMU, Imphal – 795001, India.

Nahakpam Daniel Mangang

Department of Mathematics, DMU, Imphal – 795001, India.

Khundrakpam Saratchandra Singh

Department of Physics, D. J. College, Baraut – 250611, India.

*Author to whom correspondence should be addressed.


An enhanced technique of constructing multiple odd magic square matrices is proposed in this work. A specific rule of establishing improved odd magic to magic squares derived from the odd algebraic Latin squares is studied and programmed here. Magic squares are practically important from the properties of their equality in the sum of rows, columns and diagonals. An n x n  odd magic square is an array containing the positive integers from 1 to n2, arranged so that each of the rows, columns, and the two principal diagonals have the same sum.

Keywords: Latin squares, magic squares, pivot element, vertical and horizontal pivot elements, OMS, elliptic curves

How to Cite

Singh , S. S., Singh , N. A., Mangang , N. D., & Singh , K. S. (2024). Enhanced Technique of Constructing Multiple ODD Magic Square Matrices. Asian Journal of Mathematics and Computer Research, 31(2), 29–39. https://doi.org/10.56557/ajomcor/2024/v31i28637


Download data is not yet available.


Available:http://en.wikepedia.org/wiki/Magic_squares, pp. 1-3.

Pappas T. The Joy of Mathematics. World Publishing, U.S.; New edition, [23 Jan 1993].

Boyer CB. And revised by Uta C. Merzbach: A History of Mathematics. Wiley; Revised Edition; 1998.

D. E. Smith: History of Mathematics Volume II. Dover Publications, Inc, 180 Varick Street, New York – 10014, First Edition; 1958.

Ganapathy G, Mani K. Add-on security model for public-key cryptosystem based on magic square implementation. Proceedings of the World Congress on Engineering and Computer Science 2009; Vol I., WCECS, San Francisco, USA; 2009.

David M Burton. The history of Mathematics. (3rd ed), McGraw-Hill; 1997.

Arthur Benjamin, Kan Yasuda. Magic Squares Indeed! Amer. Math. Monthly. 1999;152-156:106.

Martin Gardner. Penrose Tiles to Trapdoor Codes … and The Return of Dr. Matrix; W.H. Freeman; 1989.

Salam S, Longjam J, Moirangthem S. Encryption technique of concealing highly explosive chemicals with multiple odd magic square constructions. Quest Journal of Research in Applied Mathematics. 2023;9(3):22-31.

Tomba I. A technique for constructing odd-order magic squares using basic Latin squares. International Journal of Scientific and Research Publications. 2012;2:550-554.

Salam S, Moirangthem S. Construction of multiple odd magic squares. Asian Journal of Mathematics & Computer Research. 2022;29(1):42-52.

Abe G. Unsolved Problems on Magic Squares. Disc. Math. 1994;127:3-13.

Su Francis E, et al. Making magic squares. Math Fun Facts Available:<http://www.math.hmc.edu/funfacts>

Adam Rogers and Peter Loly: The Inertial Properties of Magic Squares and Cubes. 2004;1-3.

Available:http://mathworld.wolfram.com, pp. 1-3.

Pickover CA. The zen of magic squares, circles and stars. An Exhibition of Surprising Structures Across Dimensions; NJ: Princeton University Press; 2004.

Ezra Brown. Magic squares, finite planes, and points of inflection on elliptic curves. 2018;260-267:30.

Kaul BL, Singh R. Generalization of magic square (numerical logic) 3 ×3 and its multiples (3 ×3) ×(3 ×3), Int. J. Intell.Syst.Appl. 2013;1:90–96.

Kraitchik M. Magic squares. Ch.7 in Mathematical Recreations, New York, Norton. 1942;142-192.

Loly PD. The invariance of the moment of inertia of Magic Squares. The Mathematical gazette; 2004.