,
where 
is said to be the Order
of the Magic Square. For example, if we have the following Magic Square:Magic Squares
Date:- 20 Feb 2000.
My interest in Magic Squares:
Before going into the definition of Magic Squares, I shall like to say how I got interested into the subject. It all started one boring Saturday afternoon, back in October 1997, when final year BSc. Mathematics students at Fergusson College, University of Pune [where I used to study], were supposed to attend "The Saturday Club". There, we would listen to a guest-lecturer, invited from another University in India, who would deliver a lecture on a mathematical topic.
So, it was one of those Saturdays when, still full up with my late lunch and feeling sleepy [as usual !!!!], I attended that lecture on Magic Square which would forever remain printed in my memory as one of the best lectures I have attended in my life.
I must also add that from that day, I started doing some reading on Magic Square and that brought me to read about one of the greatest [if not the greatest] Indian mathematician Srinivas Ramanujan [1887-1920]. He did enormous amount of work on Magic Squares, more precisely how to solve them and this at a tender age of 11.
The bits that follow here are merely a copy of the work done by this great man and parts which I could withdraw from Eric Weisstein's Encyclopedia. I am still a novice in the subject, and I shall appreciate any comments and ideas for improvements. My email address is given on the left.
Magic Squares:
A (normal) magic square consists of the distinct positive
integers 1, 2, 3, ...,
where is said to be the Order
of the Magic Square. For example, if we have the following Magic Square:
We shall call this a 3*3 Magic Square (i.e
, where
= 3 here). The Order is 3. For
obvious reason, we can insert 9 integers (numbers 1 to 9, i.e 1 to
) in the Magic Square shown
above.
More explicitly, Order
means that there are
numbers in any horizontal,
vertical or "main" diagonal line. The aim here is to insert those
numbers (i.e 1 to
) in such a way that the sum
in any line (horizontal, vertical or "main" diagonal) is the same. This sum is
known as the Magic Number or the Magic Constant, and can be
calculated as:
Magic Number =
[------- Equation 1.]
Hence, for a magic number of Order 3, we expect that the
Magic Number to be 15 (Check this by replacing
by 3 in Equation 1.).
It is interesting to note the following couple of points:-
Proof by Contradiction:-
Suppose such 2*2 Magic Square did exist. Then we would have had a Magic Square in the following form,
|
a |
b |
|
c |
d |
Where a, b, c and d can take distinct values from 1 to 4. Now, since the sum in any line must be the same, the following equation should stand:-
a+b = b+d
which clearly gives a = d, hence a contradiction to the fact that a, b, c and d are distinct.
Therefore, a Magic Square of Order 2 does not exist.
----------!!!End of Proof!!!----------
So quite conveniently, the simplest Magic Square which exists is the one with Order 3; and it is given as follows:-
|
8 |
1 |
6 |
|
3 |
5 |
7 |
|
4 |
9 |
2 |
And, as we can check from Equation 1., the Magic
Number is 15 [Here, = 3].
Magic Squares of Order 4 through 8 are shown below:-
4*4
|
16 |
2 |
3 |
13 |
|
5 |
11 |
10 |
8 |
|
9 |
7 |
6 |
12 |
|
4 |
14 |
15 |
1 |
Magic Number = 34
5*5
|
19 |
21 |
3 |
10 |
12 |
|
25 |
2 |
9 |
11 |
18 |
|
1 |
8 |
15 |
17 |
24 |
|
7 |
14 |
16 |
23 |
5 |
|
13 |
20 |
22 |
4 |
6 |
Magic Number = 65
6*6
|
32 |
29 |
4 |
1 |
24 |
21 |
|
30 |
31 |
2 |
3 |
22 |
23 |
|
12 |
9 |
17 |
20 |
28 |
25 |
|
10 |
11 |
18 |
19 |
26 |
27 |
|
13 |
16 |
36 |
33 |
5 |
8 |
|
14 |
15 |
34 |
35 |
6 |
7 |
Magic Number = 111
7*7
|
48 |
22 |
3 |
33 |
14 |
37 |
18 |
|
10 |
40 |
21 |
44 |
25 |
6 |
29 |
|
28 |
2 |
32 |
13 |
36 |
17 |
47 |
|
39 |
20 |
43 |
24 |
5 |
35 |
9 |
|
1 |
31 |
12 |
42 |
16 |
46 |
27 |
|
19 |
49 |
23 |
4 |
34 |
8 |
38 |
|
30 |
11 |
41 |
15 |
45 |
26 |
7 |
Magic Number = 175
8*8
|
64 |
2 |
3 |
61 |
60 |
6 |
7 |
57 |
|
9 |
55 |
54 |
12 |
13 |
51 |
50 |
16 |
|
17 |
47 |
46 |
20 |
21 |
43 |
42 |
24 |
|
40 |
26 |
27 |
37 |
36 |
30 |
31 |
33 |
|
32 |
34 |
35 |
29 |
28 |
38 |
39 |
25 |
|
41 |
23 |
22 |
44 |
45 |
19 |
18 |
48 |
|
49 |
15 |
14 |
52 |
53 |
11 |
10 |
56 |
|
8 |
58 |
59 |
5 |
4 |
62 |
63 |
1 |
Magic Number = 260
I should leave it to the reader to discover the trend used to solve the Magic Squares. Please notice that it is easier to follow the trend of odd Order (e.g Order 3, 5, 7..) and even Order (e.g 4, 6 , 8..) Magic Square separately.
As a hint, I should draw the readers' attention to the odd Order magic Squares, that the Order 3 and Order 5 Magic Squares have been solved using a specific trend while Order 7 Magic Square has been dealt with a different trend. The reader should be convinced that both trends are very effective to solve higher odd Order magic Square.
As far as the Magic Squares of even Orders are concerned, the trends are certainly more interesting to follow. Without going into further details, the reader is advised to analyze the similarity between Magic Square of Order 4 and that of Order 8. Similarly, the Magic Square of Order 6 has an interesting behaviour imaged on the Magic Square of Order 3.
I'll leave the reader to discover these trends and I shall appreciate suggestions and questions to be forwarded to me as far as these magic Squares are concerned.
Is the Magic Square for a specific Order unique ?
Absolutely not!! Another way of getting another Magic Square of the same Order (with obviously same Magic Number) is as follows:-
If every number in a Magic Square is subtracted from
+1, another Magic Square is
obtained. This Magic Square is called the Complementary Magic Square.
Conclusion:-
This was just a brief explanation about Magic Square. The work does not stop here. Magic Square can be extended to:-
These are squares which are magic under multiplication instead of addition.
These are squares which are magic under both multiplication and addition.
If replacing each number by its square in a Magic Square produces another Magic Square, the square is said to be a bimagic square.
Reference:
© Bheshaj Kumar Ashley Hoolash