Fun with numbers – Magic Squares
The sweeter side of the MATHS…
Magic squares are a set of numbers, arranged in a set of 3 x 3 or 4 x
4 or 5 x 5 squares, in which all the Vertical, Horizontal and both diagonal totals
are same, called as the “Magic Sum”.
Normally
in a 3 x 3 magic square natural numbers 1 to 9 are used, and for 4 x 4 numbers
1 to 16 are to be used. In general, for
a “n x n” magic square numbers, “1 to n2”
numbers are to be used. And the “Magic
Sum” of the magic square must be {(1 + n 2)/2}* n. (First number plus last number to be added, and the sum divided
by two, and then multiplied by the number of rows). These magic squares can be
done from 3 x 3, 4 x 4… up to any highest number in the world.
Magic
squares can also be made with using fractions, zeros, negative numbers, with
squares and cubes. We can make magic squares for a given numbers, or size, or for
a given date or date of birth also.
For 3 x 3 magic
square, numbers 1 to 9 are to be used.
The row total will
be {(1 + n 2)/2}* n ={(1 + 9)/ 2}*3 = 5 x 3 = 15.
3 x 3 Magic
square (Total 15) 3 x 3 (Total 2019)
|
8 |
1 |
6 |
|
3 |
5 |
7 |
|
4 |
9 |
2 |
|
676 |
669 |
674 |
|
671 |
673 |
675 |
|
672 |
677 |
670 |
{(1+ 16) /2} x 4 = 34 4
x 4 (Total 2018)
|
16 |
2 |
3 |
13 |
|
5 |
11 |
10 |
8 |
|
9 |
7 |
6 |
12 |
|
4 |
14 |
15 |
1 |
|
512 |
498 |
499 |
509 |
|
501 |
507 |
506 |
504 |
|
505 |
503 |
502 |
508 |
|
500 |
510 |
511 |
497 |
6 x 6 Magic Square
(Total 111)
|
35 |
1 |
6 |
26 |
19 |
24 |
|
3 |
32 |
7 |
21 |
23 |
25 |
|
31 |
9 |
2 |
22 |
27 |
20 |
|
8 |
28 |
33 |
17 |
10 |
15 |
|
30 |
5 |
34 |
12 |
14 |
16 |
|
4 |
36 |
29 |
13 |
18 |
11 |
There are many interesting magic squares to enjoy:
Smallest Prime number magic square: All numbers in the magic square are prime numbers
(Total 177)
|
71 |
89 |
17 |
|
5 |
59 |
113 |
|
101 |
29 |
47 |
Reverse Magic squares:
All numbers in a magic square are written from the reverse and that also form a magic square and with the same total.
(Total
1515) (Total 1515)
|
806 |
107 |
602 |
|
301 |
505 |
709 |
|
408 |
903 |
204 |
|
608 |
701 |
206 |
|
103 |
505 |
907 |
|
804 |
309 |
402 |
Upside-down magic square:
A magic square when turned to 1800 also giving the same total 24.
(Total 24)
|
9 |
1 |
8 |
0 |
6 |
|
0 |
6 |
9 |
1 |
8 |
|
1 |
8 |
0 |
6 |
9 |
|
6 |
9 |
1 |
8 |
0 |
|
8 |
0 |
6 |
9 |
1 |
This article is written by Mr. T.R.
Jothilingam, Retd. Rly. Station Supt., Madurai, He is a poly mathematician and magic square wizard with several record in his name. He is also a winner of Ramanujan Award of All India Ramanujan Maths Club, Gujarat
Contact him at :- +91 94428
10486
email: mailtojothi@gmail.com
