Magic circle (mathematics)

Yang Hui's Magic concentric Circles

Magic circles were invented by the Song dynasty (9601279) Chinese mathematician Yang Hui (c. 12381298). It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameter are identical. One of his magic circles was constructed from 33 natural numbers from 1 to 33 arranged on four concentric circles, with 9 at the center.

Yang Hui magic circles

Yang Hui's magic circle series was published in his Xugu Zhaiqi Suanfa《續古摘奇算法》 (Sequel to Excerpts of Mathematical Wonders) of 1275. His magic circle series includes: magic 5 circles in square, 6 circles in ring, magic eight circle in square magic concentric circles, magic 9 circles in square.

Yang Hui magic concentric circle

Yang Hui's magic concentric circle has the following properties

Yang Hui magic eight circles in a square

Yang Hui 8 magic circles in a square 八阵图

64 numbers arrange in circles of eight numbers, total sum 2080, horizontal / vertical sum =260.

From NW corner clockwise direction, the sum of 8-number circles are:

40+	24+	9+	56+	41+	25+	8+	57	=		260

14+	51+	46+	30+	3+	62+	35+	19	=		260

45+	29+	4+	61+	36+ 20+	13+	52		=		260

37+	21+	12+	53+	44+	28+	5+	60	=		260

47+	31+	2+	63+	34+	18+	15+	50	=		260

7+    	58+	39+	23+	10+	55+	42+	26	=		260

38+	22+	11+	54+	43+	27+	6+	59	=		260

48+	32+	1+	64+	33+	17+	16+	49	=		260

Also the sum of the eight numbers along the WE/NS axis

14+	51+	62+	3+	7+	58+	55+	10	=		260

49+	16+	1+	64+	60+	5+	12+	53	=		260

Furthermore, the sum of the 16 numbers along the two diagonals equals to 2 times 260: 40+	57+	41+	56+	50+	47+	34+	63 +	29+	4+	13+	20+	22+	11+	6+	27=2*260=520

Yang Hui Magic Nine circles in a square

Yang Hui 9 magic circles in a square 连环图

72 number from 1 to 72, arranged in nine circles of eight number circle in a square; with neighbouring numbers also forming four additional 8-number circles:

form out of the borders of the following 8-circles:

(NW,N,W,C)
(NE,N,E,C)
(SW,S,W,C)
(SE,S,E,C)

thus making a total of 13 8-circles in a square:

NW,N,NE,E,SE,S,SW,W,C(center),(NW,N,W,C),(NE,N,E,C),(SW,S,W,C),(SE,S,E,C)

Ding Yidong magic circles

Ding Yidong magic circles

Ding Yidong was a mathematician contemporary with Yang Hui, in his 6th order magic circle with 6 rings, the 5 out rings have connection with a 3rd order magic square: the unit number of the 8 numbers on any ring form a 3rd order magic square.

4 9 2
3 5 7
8 1 6

Method of construction:

Let radial group 1 =1,11,21,31,41
Let radial group 2=2,12,22,32,42
Let radial group 3=3,13,23,33,43
Let radial group 4=4,14,24,34,44
Let radial group 6=6,16,26,36,46
Let radial group 7=7,17,27,37,47
Let radial group 8=8,18,28,38,48
Let radial group 9=9,19,29,39,49
Let center group =5,15,25,35,45

Arrange group 1,2,3,4,6,7,9 radially such that

2 position etc.

number 5 on group 1 radial
number 10 on group 2 radial
number 15 on group 3 radial

...

number 45 on group 9 radial

Cheng Dawei magic circles

Cheng Dawei, a mathematician in the Ming dynasty, in his book Suanfa Tongzong listed several magic circles

References

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