A.C.G.Perera and W.V.Nishadi / Elixir Appl. Math.132 (2019) 53349-53351 53349
Introduction
The name “Magic Square” was inspired by the ancient
Chinese literature where it was named as the Lo Shu square.
It was told to have been painted on the shell of a sea turtle
and was a special type of Latin square. For centuries, math
enthusiasts and mathematicians alike have found the
recreation mathematical topic of magic squares in nature to
be an interesting and entertaining subject such as rhythm in
music and very popular puzzle Soduku. [1, 6]
Definition (Latin Squares)
A Latin square is an array filled with different
symbols, each occurring exactly once in each row and exactly
once in each column. An example of a 3x3 Latin square is:
(Fig. 1)
1 3 2
3 2 1
2 1 3
1. Latin square of order 3.
Definition (Reduced Latin Squares)
It is said to be reduced (also, normalized or in standard
form) if both its first row and its first column are in their
natural order. (Fig. 2)
1 2 3
2 3 1
3 1 2
Figure 2.Reduced Latin square of order 3.
Definition (Magic Squares)
A natural magic square of order is a square array of
numbers consisting of distinct positive integers
arranged such that each cell contains a different integer and
the sum of the integers in each row, column, and diagonal is
always the same number known as the magic constant or
magic sum. The magic constant of a normal magic square
with numbers depends only on and equal to
(
)
. A line
of a magic square is any row, any column or either of the two
main diagonals of the square. A magic square of order thus
has such lines. [2]
Throughout our work, we concern magic squares when
is odd. Then middle cell is
(
)
. It is called as the pivot
element. (Fig. 3)
4 9 2
3 5 7
8 1 6
Figure 3.Natural magic square of order 3.
Pivot element
(
)
and
Magic constant
(
)
Material and Methods
There is no specific method or limited algorithm to build
all types of magic squares. The present work seeks for the
method of creating magic squares of order
, when is odd
and . There is something about the symmetry and
patterns contained in such squares that carry great appeal. We
shall prove the following simple and pleasing properties
which are exhibited by any odd-order magic square. [4, 5]
Case I: The construction of magic square of odd orders by
using basic Latin square can be expressed in the following
steps:
Step1: First, arrange the reduced Latin square by using the
cyclic shifting method [3] with the entries ( ) (
) (( )
) . (Fig. 4) & (Fig. 5)
1 2 3 ... ... .... n-2 n-1 n
2 3 4 ... ... ... n-1 n 1
3 4 5 ... ... ... n 1 2
. . . . . . .
. . . . . . .
. . . . . . .
n-
2
n-
1
n ... ... ... n-5 n-4 n-3
n-
1
n 1 ... ... ... n-4 n-3 n-2
n 1 2 ... ... ... n-3 n-2 n-1
Figure 4. Reduced Latin square with entries 1, 2 ,...,n.
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© 2019 Elixir All rights reserved
ARTICLE INFO
Article history:
Received: 9 June 2019;
Received in revised form:
29 June 2019;
Accepted: 9 July 2019;
Keywords
Latin Squares,
Reduced Latin Squares,
Magic Squares,
Cyclic Shifting.
Construction of Magic Squares of Orders
Where is Odd and
A.C.G.Perera
1
and W.V.Nishadi
2
1
Faculty of Engineering Technology, The Open University of Sri Lanka, Nawala, Sri Lanka.
2
Department of Mathematics, Faculty of Science, University of Peradeniya, Sri Lanka.
ABSTRACT
Present paper is an important study for constructing magic squares of odd orders. This
magic squares have been formed using the properties of Latin squares. Here, r educed
Latin square is used and it is formulated by using the cyclic shifting method with the
entries ( ) ( ) (( )
).This work has been
generalized for magic square of order
, when is odd and by using the
mathematical formula ≡ (
):
. The method is tested for
and .
© 2019 Elixir All rights reserved.
Elixir Appl. Math.132 (2019) 53347-53349
Applied Mathematics
Available online at www.elixirpublishers.com (Elixir International Journal)