Yang Hui's magic circle series was published in his Xugu Zhaiqi Suanfa《續古摘奇算法》 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
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: Also the sum of the eight numbers along the WE/NS axis Furthermore, the sum of the 16 numbers along the two diagonals equals to 2 times 260:
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: 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.
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: Method of construction: 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
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.
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.
