MS1. MAGIC SQUARES

Today I will introduce you to MAGIC SQUARES

In an arrangement of n horizontal rows and n vertical columns there will be nxn cells which can be filled by nxn numbers in such a way that they satisfy a set of specified conditions. For example we may require that the sum of all the rows are equal to each other or the sum of each vertical column is not only equal to each other, but also equal to the sum of each row or we may specify that only odd numbers are used to give a specified total, etc..
If we go a step further and specify that not only the sum of each row, each column and the main diagonals are equal but also that, only nxn consecutive numbers are used, the resultant square pattern has been called, from ancient times, a MAGIC SQUARE. As such a magic square of size n, denoted by Sn, is an nxn square which can be said to have the following properties:-
i) It has nxn consecutive integers, each integer occurring only once.
ii) The total for each row, each column and the main diagonals of the square is the same. This is the equisum property. The total, so obtained, is called the magic sum of the square.
(a) If these integers are 1,2,3,...n^2, the sum is n(n^2 +1)/2.
(b) In general, if the first integer is m+1, the sum is mxn + n(n^2+1)/2. Without loss of generality we may assume that the first (lowest) integer of Sn is 1. To get a square in which the sequence of integers begins with some other number m+1, we add m to all the cells in Sn.
(c) If n is larger than 4, there are a large number of unique solutions for Sn.
An example of S4 with magic sum of 34 is given below:-
01 13 12 08
06 10 03 15
11 07 14 02
16 04 05 09

MS2. Magic Squares In Ancient Times.
The construction of magic squares is an amusement of great antiquity, we hear of magic squares in China and India before the Christian era, while they appear to have been introduced to Europe by Moscowpulus in Constantinople in early 15th century. However, what was at first merely a practice of magicians and talisman makers, has now, for a long time, become a serious study for mathematicians. Not that they imagined that it would lead them to anything of solid advantage, but because the theory was seen to be fraught with difficulty and it was considered that some new properties of numbers might be discovered which mathematicians could turn to account. This has, in fact, proved to be the case. For, from a certain point of view, the subject has been found to be algebraic rather than arithmetical and to be intimately connected with great departments of science, such as, the infinitesimal calculus, the calculus of operations and the theory of groups.
Till the advent of modern computers, it is understood, that, no living person knew, in how many ways, it is possible to form a square Sn of order exceeding 4. Even now it is not known how many possible squares are there of order 6 and above.
Magic Squares In China.
The Chinese appear to have invented and used the Magic Squares since long time back; they are mentioned in a Chinese book written four to five thousands years before our era. The Worlds oldest square, a Chinese creation, is reproduced below:-
It is known as the LO SHU.
4 9 2
3 5 7
8 1 6