© 2017 Xiaoyang Ma et al., published by De Gruyter Open.
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Spec. Matrices 2017; 5:82–96
Research Article Open Access
Xiaoyang Ma*, Kai-tai Fang, and Yu hui Deng
Some results on magic squares based on
generating magic vectors and R-C similar
transformations
DOI 10.1515/spma-2017-0009
Received April 10, 2016; accepted April 28, 2017
Abstract: In this paper we propose a new method, based on R-C similar transformation method, to study
classification for the magic squares of order 5. The R-C similar transformation is defined by exchanging two
rows and related two columns of a magic square. Many new results for classification of the magic squares of
order 5 are obtained by the R-C similar transformation method. Relationships between basic forms and R-C
similar magic squares are discussed. We also propose a so called GMV (generating magic vector) class set
method for classification of magic squares of order 5, presenting 42 categories in total.
Keywords: Generating magic vector, Magic squares, Similar matrix.
1 Introduction
A (classical) semi-magic square of order n is an n × n matrix formed by n
2
consecutive integers,
{1, 2, ... , n
2
}, such that the sum of each row and each column are the same. This sum is called magic sum
and denoted by µ
n
. It is easy to see that µ
n
= n(n
2
+ 1)/2. For example, µ
3
= 15, µ
4
= 34, and µ
5
= 65.A
semi-magic square of order n is called magic square if the sums of its two diagonals are equal to µ
n
.
The first magic square
M
0
=
8 1 6
3 5 7
4 9 2
(1)
was found by a Chinese. With a history of more than 3000 years, the magic square has been an attractive topic
in mathematics and related research and applications. There is a rich literature on this interesting topic. Pick-
over [8] and Wu [10] gave a good introduction to the magic square and its development. However, the magic
square is still mysterious and there are a lot of open questions to be solved. Classification is an important one
of them. A lot of researchers such as Ollerenshaw and Bondi [7], Chu and Styan [2], and Fang, Luo and Zheng
[3] studied the classifications of magic squares of order 3 and 4.
Denote the set of magic squares of order n by M
n
. When the order n increases the number of magic squares
in M
n
is exponentially growing. For example, the amount of magic squares is 7040 in M
4
and 2,202,441,792
in M
5
. As a consequence, there are much less publications on classification for the magic squares of order 5.
For a related article, the readers can refer to Candy [1].
Now let us review some useful concepts for classification.
*Corresponding Author: Xiaoyang Ma: Division of Science and Technology, BNU-HKBU United International College, Zhuhai,
China; Department of Biostatistics, Georgetown University, Washington D.C., USA, E-mail: lnnay@qq.com
Kai-tai Fang: Division of Science and Technology, BNU-HKBU United International College, Zhuhai, China; The Key Lab of
Random Complex Structures and Data Analysis, The Chinese Academy of Sciences, Beijing, China; E-mail: ktfang@uic.edu.hk
Yu hui Deng: Division of Science and Technology, BNU-HKBU United International College, Zhuhai, China, E-mail:
ivandeng@uic.edu.hk