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Rene Descartes
“'Cogito Ergo Sum' means 'I think, therefore I am'. ”
 
 
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About Magic Square Game
   

T.N. Mahesh (LIC DEVELOPMENT OFFICER)
Game Inventor

Welcome to Magic Square web site dedicated to the puzzles in the magic squares.

First let us see what a magic square is. 

A Magic Square is an arrangement of numbers from 1 to n2 in an n x n matrix. All the numbers should occur only once. The sum of the entries of any row, column, or any main diagonal is the same. This is called the magic sum.

The magic sum should be n(n^2 + 1)/2.

The simplest magic square is the order 1 magic square.

1
Table 1

This is also called the God magic square.

There is no magic square in order 2x2.

There is only one magic square in order 3x3.

15
8
1
6
3
5
7
4
9
2
Table 2

Look at the above square. This is the only square in the order 3x3. This same square could be transposed or its mirror image could be drawn like

15
8
3
4
1
5
9
6
7
2
Table 3
Or
15
2
9
4
7
5
3
6
1
8
Table 4

All these are same.

In the magic square the rotation, mirror image or transpose are not counted. They are treated the one and the same.

So therefore there is only one magic square of the order 3x3.

This magic square is also called the Seetha Chakra as per the Hindu almanacs.

When a magic square is dedicated to the name of Seetha, then surely a magic square dedicated to the name of her husband Sri Rama should also be there, isn’t it?

Yes you are right. There is one square by the name of Sri Rama Chakra. This is a 4x4 magic square.

Before we go deep into the Sri Rama Chakra let us see what a 4x4 magic square is.

Typically a 4x4 magic square consists of 1 to 16 numbers. These are arranged in such a way that all their rows, columns and their two main diagonals give the magic sum of 34.

Bessy of France found out that there are about 880 different 4x4 magic squares.

Dudeney found out that these 880 different magic squares of order 4x4 can be divided into 12 groups.

We will see two examples of this 4x4 magic square.

The first one is called the Durer magic square.

34
16
3
2
13
5
10
11
8
9
6
7
12
4
15
14
1
Table 5



This was discovered by Albrecht Durer in 1514. The Bessy number for this is 112 and it belongs to the group III of the Dudeney’s classification.

It has a many interesting properties.

(Insert here the image of the melancholia as found in the wikipaedia).

The next square that we will see here is the Jaina Square. This is found in the temple of Khajuraho in India.

34
7
12
1
14
2
13
8
11
16
3
10
5
9
36
15
4
Table 6

The last one we will see is the Sri Rama Chakra.

34
9
16
5
4
7
2
11
14
12
13
8
1
6
3
10
15
Table 7

Both the Sri Rama Chakra and the Seetha Chakra are found in the Hindu almanacs.

Both the Jaina square and the Sri Rama Chakra belong to the group I classification of the Dudeney.

My Mother was fascinated with the Sri Rama Chakra by looking at the Hindu almanac.

She thought why this 4x4 matrix is named after a Hindu God.

This magic square was used to predict one’s future in the Hindu almanacs. That comes under the philosophy of the magic squares. That is an interesting topic. It would imply that the good and the bad co–exist.

Whichever way you start from you reach the Ultimate. This life is a zero sum game.

That is a wonderful philosophy. The Hindu almanac was using these Sri Rama Chakra and the Seetha Chakra only for that purpose.

Evidently they are genius and they must have known more about the whole field of the magic squares.

They didn’t reveal anything to the modern world explicitly.

The rare genius of the Hindu is to apply all the knowledge and science at his command to living a contended and God centric life.

They used the science of the magic squares for the very same purpose.

My Mother was only a Matriculate and she didn’t know anything about magic squares before.

She kept thinking why this 4 x 4 matrix was named after a Hindu God. This thought kept her engaged. This Sri Rama Chakra beckoned her again and again and it was throwing this challenge to her.

On close observation she found that no number is repeated and that all the row, column and the two main diagonals add up to 34.

Then she knew that it was a magic square.

Then she wanted to construct the magic square for any magic sum (based on the Sri Rama chakra).

She found a way to do the same.

Still the Sri Rama Chakra was teasing her with a question ‘can you scale me up?’

She wanted to construct an 8 x 8 magic square based upon the pattern of the Sri Rama Chakra.

Hindu women have the peculiar habit of drawing geometric figures using dots and lines in front of their houses every day. This is called Kolam in Tamil and Rangoli in Hindi.

From a small kolam she would want to draw a bigger kolam.

Likewise she wanted to scale up the 4 x 4 into an 8 x 8 based upon the Sri Rama Chakra.

She spent many years on this project and then found out a way to do the same.

Initially she didn’t think her work was unique. She thought that some one could have already done it. You can call that modesty. Yes. The character of the Indians is being humbly humble.

She didn’t lay any claim to it. After all she was indulging in this hobby only to amuse herself.

Once she constructed the 8 x 8 she knew that she can construct the magic square of any dimension in the multiples of 4 for any sum.

Then she constructed the 100 x100 and the 1000 x 1000 and she found a mention in the Limca Book of records in the year 2001.

She was honored by the Ramanujan Museum in the Royapuram in Chennai.

She was also interviewed in the Sun TV’s Saadhanai Nayagan program.

We cannot thank adequately Mr. P.K. Srinivasan, who encouraged my Mother. He also suggested to her about writing a book about her discoveries in the magic squares.

Eventually she did write a book on magic squares titled ‘MAGIC OF MAGIC SQUARES’

Sri Rama Chakra is one among the 880 different 4 x 4 magic squares.

Then I took over the work of my Mother.

I thought why only the Sri Rama Chakra alone could be scaled up. How to scale up all the 880 4 x 4 Magic Squares?

Then I found out the answer in the form of the mimic method.

This method I shall explain later. Now let us continue with our rambling on Magic Squares.

Once I could scale up any 4 x 4 into an 8 x 8 I knew that the possibilities are enormous. One peculiarity about this mimic method is that not only the 8 x 8 is a  Magic Square, also all the four 4 x 4 subsquares are Magic Squares.

Let us see this with an example.

The following 4 X 4  Magic Square :

34
1
8
10
15
14
11
5
4
7
2
16
9
12
13
3
6
Table 8

has been scaled up in to a 8 X 8 magic square as given below using the mimic method.

260
1
32
40
57
2
31
39
58
56
41
17
16
55
42
18
15
25
8
64
33
26
7
63
34
48
49
9
24
47
50
10
23
3
30
38
59
4
29
37
60
54
43
19
14
53
44
20
13
27
6
62
35
28
5
61
36
46
51
11
22
45
52
12
21
Table 9

The 8 x 8 in the above figure is a magic square.

Also all the four 4 x 4 sub squares are also magic squares.

The following is the sub square I.

130
1
32
40
57
56
41
17
16
25
8
64
33
48
49
9
24
Table 10

This is sub square II.

130
2
31
39
58
55
42
18
15
26
7
63
96
47
50
10
23
Table 11

This is sub square III

130
3
30
38
59
54
43
19
14
27
6
62
35
46
51
11
22
Table 12

And this is the sub square IV

130
4
29
37
60
53
44
20
13
28
5
61
36
45
52
12
21
Table 13

Note that all the four sub squares give the magic sum. If the magic sum of a 8 x 8 is 260 then the magic sum of the 4x4 sub square is 130.

Keeping this property in mind and also the rules for the birth day magic squares as discovered and enumerated by the great Indian Mathematician Srinivasa Ramanujan in mind I have devised the magic square puzzles.

I do not know how many 8 x 8 magic squares are possible. One very very conservative estimate tells that 6.87x10^2210 magic squares are possible in the order 8 x 8. Definitely there must be more to it than meets the eye.

I do not know how to count the number of magic squares. I am wonderstruck.

Likewise I could construct a magic square of any order in multiples of 4.

Have a nice time in the puzzle section.

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