1. Introduction

Magic squares generally fall into the realm of recreational mathematics (Pasles, 2008), (Pickover, 2002) however a few times in the past century and more recently, they have become the interest of more-serious mathematicians. Magic squares have spelt fascination to mankind throughout history and all across the globe. A normal magic square is a square array of consecutive numbers from 1…n^2 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 of order 4 have 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. The study on numerical properties of strongly magic squares of order 4 have been carried out by astrologer turned mathematician Padmakumar (Padmakumar, 1995). Another study carried out by Stanley [8] on magic Squares using the tools of Commutative Algebra which makes use of graded rings to define a hilbert series (Qimh Richey Xantcha, 2012). The homomorphic and isomorphic properties on semi magic squares has also studied recently (Sreeranjini, 2014). In this paper some advanced mathematical properties of the strongly magic squares are discussed.

2. 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 n^th order matrix with entries in such that adhere to this paper in appearance as closely as possible.

(1)

(2)

(3)

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 (Small, 1988).

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 In the example given below the magic constant of A is 15 and B is 34.

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

(4)

(5)

(6)

(7)

where the subscripts are congruent modulo

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 k column gaps or k row gaps is generally denoted as or respectively.

2.4. Group Homomorphism

A mapping from a group into a group is a homomorphism of into if

for all [9]

2.5. Group Isomorphism

A one to one onto homomorphism from a group into a group is defined as isomorphism (Fraleigh, 2003).

2.6. A One to One and onto Mapping

A function is one to one if only when The function is onto of Y if the range of is Y.

2.7. Kernel of a Homomorphism

If is a homomorphism of a group into then the kernel of is denoted as and is defined as where is the identity of

2.8. Linear Transformation

Let and be two vector spaces over the same field Then a mapping is called linear transformation of into if

and (Kenneth Hoffmann, 1971).

2.9. Other Notations

1. denotes the set of all real numbers.

2. denote the set of all strongly magic squares of order

3. denote the set of all strongly magic squares of order

denote the set of all strongly magic squares of the form such that for every Here A is denoted as i.e. If then

4. denote the set of all strongly magic squares of order with magic constant 0, i.e. If then

3. Propositions and Theorems

Proposition 3.1

If and are two Strongly magic squares of order with and , then is also a Strongly magic square with magic constant for every

Proof:

Let and

Then

Sum of the i^th row elements of

A similar computation holds for column sum, diagonals sum and sum of the sub squares.

From the above propositions the following results can be obtained by putting suitable values for

Results:

If for every

Theorem 3.2

forms an abelian group.

Proof:

I. Closure property: if then (from above result 1.2)

II. Associativity: if then (Since matrix addition is associative.)

III. Existence of Identity: There exists 0 matrix in so that where 0 acts as the identity element.

IV. Existence of additive inverse: For every there exists so that where (from result 1.5).

V. Commutativity: If then (Since matrix addition is commutative.)

This completes the proof.

Proposition 3.3

forms a subgroup of the abelian group

Proof:

It is clear that

For and then clearly

Thus forms a subgroup of the abelian group

Proposition 3.4

forms a subgroup of the abelian group

Proof:

It is clear that

Take

Now

Therefore

Thus forms a subgroup of the abelian group (Mallayya, Neeradha, 2016).

Proposition 3.5

For all

Proof:

Since

Theorem 3.6

forms a vector space over the field of real numbers.

Proof:

It is an immediate consequence of Theorem 3.2 and Proposition 3.5

Theorem 3.7

forms a vector space over the field of real numbers.

Proof:

Since and is a vector space over the field of real numbers with respect to the addition of matrices as addition of vectors and multiplication of a matrix by a scalar as scalar multiplication, it is enough to show that is a subspace of

This can be verified by the fact; for every and

Theorem 3.8

forms a vector space over the field of real numbers.

Proof:

Proceeding as in Proposition 3.7 it is enough to show that

for every

Since

Now (From result 1.4)

Thus (Neeradha. C. K, V. Madhukar. Mallayya, 2016).

Proposition 3.9

The mapping defined by is a group homomorphism.

Proof:

Let then

(By Result 1.2)

(Neeradha, Mallayya, 2016)

Proposition 3.10

The mapping defined by is a linear transformation

Proof:

Let

(By Result 1.4 and Theorem 3.6)

Proposition 3.11

The mapping defined by is a linear transformation.

Proof:

Let then such that and

From Result 1.4 and Theorem 3.7

Hence is a linear transformation.

Proposition 3.12

The mapping defined by linear transformation.

Let then and

(By Result 1.4 and Theorem 3.8)

Hence is a linear transformation.

Proposition 3.13

The kernel of the mapping defined by is Ker where and such that

Proof:

Let

Now

Therefore

Now let then

Clearly

Therefore

Theorem 3.14

The mapping defined by is a vector space isomorphism.

Proof:

Let then and

To show that is 1-1

To show that is onto

For every there exists such that

Since forms a vector space (from Theorem 3.7) and from the above shown results, the mapping

defined by is a vector space isomorphism.

4. Conclusions

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 namely Abelian group structure, vector spaces, group homomorphism, group isomorphism, vector space isomorphism, linear transformation, kernel of transformation are described. Physical application of magic squares is still a new topic that needs to be explored more. Ollerenshaw and BrÈe (Ollerenshaw, 1999) have a patent for using most-perfect magic squares for cryptography, and Besslich (Besslich, 1983), (Besslich, Ph. W, 1983) has proposed using pan diagonal magic squares as dither matrices for image processing. Further studies are being carried out by the authors on the scope for further research and the application of Strongly Magic Squares on Diophantine equations, Moment of inertia, Electric Quadrapoles, Data hiding Schemes etc.

ACKNOWLEDGEMENTS

We express sincere gratitude for the valuable suggestions given by Dr. Ramaswamy Iyer, Former Professor in Chemistry, Mar Ivanios College, Trivandrum, in preparing this paper.