# Magic circle (mathematics)

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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

• The sum of the numbers on four diameters = 147,
• 28 + 5 + 11 + 25 + 9 + 7 + 19 + 31 + 12 = 147
• The sum of 8 numbers plus 9 at the center =147;
• 28 + 27 + 20 + 33 + 12 + 4 + 6 + 8 + 9 = 147
• The sum of eight radius without 9 =magic number 69: such as 27 + 15 + 3 + 24 = 69
• The sum of all numbers on each circle (not including 9) = 2 × 69
• There exist 8 semicircles, where the sum of numbers = magic number 69; there are 16 line segments (semicircles and radii) with magic number 69, more than a 6 order magic square with only 12 magic numbers.

### Yang Hui magic eight circles in a square

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

72 number 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

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 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 magic circles

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

## Extension to higher dimensions

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]

## Relationship with magic squares

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]

 6 15 4 9 3 10 5 16 13 8 11 2 12 1 14 7

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In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as 3 × 3.

28 (twenty-eight) is the natural number following 27 and preceding 29.

In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal to the same number, the so-called magic constant of the cube, denoted M3(n). It can be shown that if a magic cube consists of the numbers 1, 2, ..., n3, then it has magic constant

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A power of two is a number of the form 2n where n is an integer, that is, the result of exponentiation with number two as the base and integer n as the exponent.

666 is the natural number following 665 and preceding 667.

800 is the natural number following 799 and preceding 801.

A pandiagonal magic square or panmagic square is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.

In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an n × n grid.

The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. In general where is the side length of the square.

260 is the natural number following 259 and preceding 261.

A centered cube number is a centered figurate number that counts the number of points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with i2 points on the square faces of the ith layer. Equivalently, it is the number of points in a body-centered cubic pattern within a cube that has n + 1 points along each of its edges.

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Yang Hui, courtesy name Qianguang (謙光), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang, Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting Yang Hui's Triangle. This triangle was the same as Pascal's Triangle, discovered by Yang's predecessor Jia Xian. Yang was also a contemporary to the other famous mathematician Qin Jiushao.

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## References

1. W. S. Andrews, MAGIC SQUARES AND CUBES, Second Edition, Revised and Enlarged, Open Court Basic Readers (1917), page 198, fig.337
• 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