MS2. MAGIC SQUARES

According to legend, the pattern was first revealed on the shell of a turtle that crawled out of the LO river in the 23rd century B.C. in the days of the legendary Emperor Yii, reputed to be a Hydraulic Engineer.
In Chinese history the pattern is considered as a mystical symbol of enormous significance. The even numbers are identified with YIN, the female principle, and the odd numbers with YANG, the male principle. The central 5 represented the earth around which, in evenly balanced Yin and Yang, were the four elements 4 & 9 symbolizing metal, 2 & 7 fire, 1 & 6 water and 3 & 8 wood.
In Chinese cosmology, from which Chinese Astrology is derived, the Positive (YANG) and Negative (YIN) aspects of reality are not seen as opposing one another, but, on the contrary, as supporting each other, like the left and the right sides of an arch. Just as by definition one arch cannot exist without having two sides, so reality cannot exist without both its Positive and Negative aspects. In the ancient Positive and Negative Taoist philosophy, everything is classified as pertaining to one of these two aspects of inter-dependent duality. Thus the Night is negative, the Day is positive; Female sexuality is negative, Maleness is positive; Sugar is negative, Salt is positive; and so on
And now have a look at how these numbers behave;-
492^2 + 357^2 + 816^2 = 438^2 + 951^2 + 276^2.

Magic Squares In India,
The order 4 magic square, with magic sum of 34, shown below on the r.h.s is over 2000 years old and comes from India, while the square on the l.h.s.is the Jaina square of Khajuraho, also from India:-
07 12 01 14 - 01 14 15 04
02 13 08 11 - 12 07 06 09
16 03 10 05 - 08 11 10 05
09 06 15 04 - 13 02 03 16

The Indian square has some interesting properties:-
1. All the four 2x2 corners and the central 2x2 square also total 34.
2. In each row one pair adds to 19 and the other to 15. In each column also one pair adds to 13 and the other to 21.
2.The squares of the numbers in the first and fourth rows are equal, as also in the second and third rows.
1^2 + 14^2 + 15^2 +4^2 = 13^2 + 2^2 + 3^2 + 16^2 = 438
12^2 + 7^2 + 6^2 + 9^2 = 8^2 + 11^2 + 10^2 + 5^2 = 310
4. Similarly the squares of the numbers in the first and fourth columns are equal, as also, in the second and third columns.
1^2 + 12^2 + 8^2 + 13^2 = 4^2 + 9^2 + 5^2 +16^2 =378
14^2 +7^2 + 11^2 + 2^2 = 15^2 + 6^2 +10^2 + 3^2 =370
Now consider the diagonals of the 2x2 squares
12 +14 +3 + 5 = 15 + 9 +8 +2 = 34 further
12^2 +14^2 + 3^2 + 5^2 =15^2 + 9^2 + 8^2 +2^2 = 374 and
12^3 +14^3 +3^3 + 5^3 = 15^3 + 9^3 + 8^3 +2^3 =4624
Similar properties are found for the Jaina square:-
1 All the four 2x2 corners and the central 2x2 square total 34.
2 In each row one pair adds to 19 and the other to 15. In each column one pair adds to 9 and the other to 25.
3 7^2+12^2+1^2+14^2 = 16^2+3^2+10^2+5^2 =390
2^2+13^2+8^2+11^2 = 9^2+ 6^2+15^2+4^2 =358
4 7^2+2^2+16^2+9^2 = 1^2+8^2+10^2+15^2 =390
12^2+13^2+3^2+6^2 = 14^2+11^2+5^2+4^2 =358
Magic Squares were constructed in India before the Christian era.