Daoism's Perfect Number Magic Square Systems
老子 數學
3, 5, 9, 15

acupuncturetoday.com
The YinYang Average
陰陽 平均份

Five basic elements of the universe:

metal 金 wood 木 earth 土 water 水 fire 火

6 ..... 1 ..... 8

7 ..... 5 ..... 3

2 ..... 9 ..... 4

the sum of three numbers 合計 = 15 ﹔ Average 平均 = 5

Horizonally 橫 : (6+1+8)=(7+5+3)=(2+9+4)= 15/3 = 5
Vertically 直 : (6+7+2)=(1+5+9)=(8+3+4) = 15/3 = 5
Diagonally 對角 : (6+5+4)=(8+5+2) = 15/3 = 5
Center 中 ﹕ (5 + 5 + 5) = 15/3 = 5
Four corner 四角 ﹕(6+2+8+4)= (1+7+9+3) = 20/4 = 5
Average 平均 = (1+2+3+4+5+6+7+8+9) / 9 = 45/9 = 5

1 -- water 水 ; 2 -- earth 土 ; 3 -- lake 湖 ; 4 -- sky 天 ;
5 -- YinYang 陰陽 ;
6 -- wind 風 ; 7 -- thunder 雷 ; 8 -- mountain 山 ; 9 -- fire 火 .

they formed FIVE complementary pairs:

1 & 9 -- Water and Fire 水 - 火
2 & 8 -- Earth and Mountain 土 - 山
3 & 7 -- Lake and Thunder 湖 - 雷
4 & 6 -- Sky and Wind 天 - 風
5 & 5 -- Yin & Yang 陰 - 陽

Generalized mathematical formula for summation of sequential integer numbers from 1 to N:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + . . . + N = N * (N+1)/2 (N times its average)

courtesy of Absolute Animation GIFs
the 2 beads on the top each represents 5; the 5 beads on the bottom each represents 1.
算盤
It was believed: the abacus, invented thousand years ago,
was based on the Daoism's number systems.

Applications of 3, 5,9,15 numbers:
3 -- in Taoism(Daoism) Verses 14, 19, 42, 50, 67: he used 3 to describe eternal life and life of all things. traditional ancestor worship with 3 pairs of chop sticks and 3 cups of wines. bow head 3 times in worship.
5 -- in verses 5, 12, Tao(Dao) is the center of the universe. (5 is the medium of the 9 numbers) The universe has 5 basic elements 金,木, 水, 火,土; 五行結構; the earth was divided into 4 directions and a center 東南西北中﹔五方說緣; 5 classes, 5 customs, 5 flavours, and 5 colors.
9 (3x3) -- Daoism text has a total of 81 verses; which is 9 times 9; 9 major planets in the solar system. 9 represents good luck number used in chinese customs, 9-coins string festival gift exchanges; and traditional formal 9 courses dinners. chinese communist party consists 9 member of the standing committee.
15(3x5) -- In abacus, one column of beads represents 15; in lunar calendar, the full moon always appear on the 15th night of each month. 15 courses in formal dinner celebration.