International Journal of Scientific and Innovative Mathematical Research (IJSIMR)
Volume 4, Issue 8, August 2016, PP 7-14
ISSN 2347-307X (Print) & ISSN 2347-3142 (Online)
DOI: http://dx.doi.org/10.20431/2347-3142.0408002
www.arcjournals.org
General Properties of Strongly Magic Squares
Neeradha. C. K. Dr. V. Madhukar Mallayya
Assistant Professor, Dept. of Science & Humanities Professor & Head of Dept. of Mathematics,
Mar Baselios College of Engineering & Mohandas College of Engineering &
Technology, Technology,
Thiruvananthapuram, Kerala, India Thiruvananthapuram, Kerala, India
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. Some interesting properties of Strongly Magic Squares are briefly described.
Keywords: Magic squares, Magic constant,Strongly Magic Squares ,Product of magic squares.
1. INTRODUCTION
Magic squares date back in the first millennium B.C.E in China [1], developed in India and Islamic
World in the first millennium C.E, and found its way to Europe in the later Middle Ages [2] and to
sub-Saharan Africa not much after [3]. Magic squares generally fall into the realm of recreational
mathematics [4, 5], however a few times in the past century and more recently, they have become the
interest of more-serious mathematicians. Srinivasa Ramanujan had contributed a lot in the field of
magic squares. Ramanujan’s work on magic squares is presented in detail in Ramanujan’s Notebooks
[6]. A normal magic square is a square array of consecutive numbers from where the rows,
columns, diagonals and co-diagonals add up to the same number. The constant sum is called magic
constant or magic number. Along with the conditions of normal magic squares, strongly magic square
has 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 [7]. There are many recreational aspects of strongly
magic squares. But, apart from the usual recreational aspects, it is found that these strongly magic
squares possess advanced mathematical properties.
2. NOTATIONS AND MATHEMATICAL PRELIMINARIES
2.1 Magic Square
A magic square of order n over a field where denotes the set of all real numbers is an nth order
matrix [ ] with entries in such that
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Neeradha. C. K. & Dr. V. Madhukar Mallayya
Equation (1) represents the row sum, equation (2) represents the column sum, equation (3) represents
the diagonal and co-diagonal sum and symbol represents the magic constant. [8]
2.2 Magic Constant
The constant in the above definition is known as the magic constant or magic number. The magic
constant of the magic square A is denoted as .
2.3 Strongly magic square (SMS): Generic Definition
A strongly magic square over a field is a matrix [ ] of order with entries in such that
Equation (4) represents the row sum, equation (5) represents the column sum, equation (6) represents
the diagonal & co-diagonal sum, equation (7) represents the sub-square sum with no gaps in
between the elements of rows or columns and is denoted as and is the magic
constant.
Note: The order sub square sum with column gaps or row gaps is generally denoted as
or respectively.
2.4 Notations
1. Z denotes the set of all positive real integers.
2. denotes the set of all real numbers.
3. SMS
3. PROPOSITIONS AND THEOREMS
Proposition 3.1
Let where be a strongly magic square of order
Proof:
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General Properties of Strongly Magic Squares
From the definition of SMS ,
Also
Thus equating the both the equations (3.1) and (3.2)
Thus
Proposition 3.2
The sum of the corner elements of a SMS is the magic constant.
Proof:
Its an immediate consequence of the definition of SMS
Proposition 3.3
Let be a strongly magic square with order and , then there exists another
strongly magic square of order with
Proof:
Let for such that
Define a square matrix in such a way that
for
Now, sum of ith row element of
since
Therefore
Similar computation holds for column elements
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Neeradha. C. K. & Dr. V. Madhukar Mallayya
For diagonal elements
Similar computation holds for co-diagonal elements
Now for the subsqaures
Proposition 3.4
Let be a SMS of order n with , then is also a SMS with where
Proof:
[ ] and
(Since a
j 1
ij )
Similarly we can calculate the sum of the column elements
For the sum of the diagonal elements;
For the sum of the co-diagonal elements;
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General Properties of Strongly Magic Squares
For the sum of the sub square elements ;
Proposition 3.5
Proof:
Let and
Sum of the ith row elements is given by
A similar computation holds for column sum also.
For the sum of the diagonal elements,
For the sum of the co-diagonal elements,
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Neeradha. C. K. & Dr. V. Madhukar Mallayya
For the sum of the sub-square elements,
Proposition 3.6
Let be a SMS of order with , then is also an SMS with
, where
Proof:
Proceeding as in Proposition 3.5, we will get the required result.
Proposition 3.7
Proof:
Let [ ] and
Sum of the ith row elements is given by
A similar computation holds for column sum also.
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General Properties of Strongly Magic Squares
For the sum of the diagonal elements,
For the sum of the co-diagonal elements,
For the sum of the sub-square elements,
Proposition 3.8
Let be a SMS of order with , then is also a SMS with
where . Here the 0 matrix is excluded.
Proof:
Proceeding as in Proposition 3.7, the required result can be obtained.
4. CONCLUSION
While magic squares are recreational in grade school, they may be treated somewhat more seriously
in different mathematical courses. The study of strongly magic squares is an emerging innovative area
in which mathematical analysis can be done. Here some advanced properties regarding strongly magic
squares are described which can be used to explore new horizons. Certainly more can be done in the
context of linear algebra.
ACKNOWLEDGEMENT
The authors express sincere gratitude for the valuable suggestions given by Dr.Ramaswamy Iyer,
Former Professor in Chemistry, Mar Ivanios College, Trivandrum, in preparing this paper.
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Neeradha. C. K. & Dr. V. Madhukar Mallayya
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,Chapter1(pp 16-24),Springer ,1985
[7] T.V. Padmakumar “Strongly Magic Square” , Applications Of Fibonacci Numbers Volume 6
[8] Proceedings of The Sixth International Research Conference on Fibonacci Numbers and Their
Applications, April 1995
[9] Charles Small, “Magic Squares Over Fields” The American Mathematical Monthly Vol. 95, No.
7 (Aug. - Sep., 1988), pp. 621-625
AUTHOR’S BIOGRAPHY
Neeradha. C. K, is working as Assistant Prof. at Mar Baselios College of
Engineering and Technology, Department of Science And Humanities, APJ Abdul
Kalam University of Technology,Kerala,India. Her fields of interest include abstact
algebra, magic squares and linear algebra.
Dr Madhukar Mallayya, is a renowned Indian Mathematician currently working as
Prof. and Head of the Department, Department of Mathematics at Mohandas College
of Engineering and Technology, APJ Abdul Kalam University of Technology,
Kerala, India. His fields of interest include numerical analysis, linear algebra and
vedic mathematics.
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