Magic circle (mathematics)

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 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 numbers from 1 to 72, arranged in nine circles of eight numbers in a square; with neighbouring numbers forming four additional eight number circles: thus making a total of 13 eight number circles:

NW		N		NE
	x1		x2
W		C		E
	x3		x4
SW		S		SE

Extra circle x1 contains numbers from circles NW, N, C, and W; x2 contains numbers from N, NE, E, and C; x3 contains numbers from W, C, S, and SW; x4 contains numbers from C, E, SE, and S.

• Total sum of 72 numbers = 2628;
• sum of numbers in any eight number circle = 292;
• sums of three circles along horizontal lines = 876;
• sum of three circles along vertical lines = 876;
• sum of three circles along the diagonals = 876.

Ding Yidong magic circles – numbers on each circle (solid colour) sum to 200 and numbers on each diameter (dashed grey) sum to 325

Ding Yidong was a mathematician contemporary with Yang Hui. In his magic circle with 6 rings, the unit numbers of the 5 outer rings, combined with the unit number of the center ring, form the following 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

• each number occupies one position on circle
• alternate the direction such that one radial has smallest number at the outside, the adjacent radial has largest number outside.
• Each group occupies the radial position corresponding to the number on the Luoshu magic square, i.e., group 1 at 1 position, group 2 at 2 position etc.
• Finally arrange center group at the center circle, such that

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

Andrews's sphere with numbers 1 to 62 arranged along intersections of 5 circles of latitude (dashed grey) and 6 circles of longitude (coloured solid)

In 1917, W. S. Andrews published an arrangement of numbers 1, 2, 3, and 62 in eleven circles of twelve numbers each on a sphere representing the parallels and meridians of the Earth, such that each circle has 12 numbers totalling 378.[1]

Magic circle derived from magic square

A magic circle can be derived from one or more magic squares by putting a number at each intersection of a circle and a spoke. Additional spokes can be added by replicating the columns of the magic square.

In the example in the figure, the following 4 × 4 most-perfect magic square was copied into the upper part of the magic circle. Each number, with 16 added, was placed at the intersection symmetric about the centre of the circles. This results in a magic circle containing numbers 1 to 32, with each circle and diameter totalling 132.[1]

• Lam Lay Yong: A Critical Study of Hang Hui Suan Fa 《杨辉算法》 Singapore University Press 1977
• Wu Wenjun (editor in chief), Grand Series of History of Chinese Mathematics, Vol 6, Part 6 Yang Hui, section 2 Magic circle (吴文俊 主编 沈康身执笔 《中国数学史大系》 第六卷 第六篇 《杨辉》 第二节 《幻圆》) ISBN 7-303-04926-6/O