Magic circle

702px-magic_circle

Magic circles were invented by the Song Dynasty (960–1279) Chinese mathematician Yang Hui (c. 1238–1298). 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 circles , with 9 at the center.

 

Yang Hui’s magic 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(semi circles and radii) with magic number 69, more than a 6 order magic square with only 12 magic numbers.

Yang Hui’s magic square was published in his Xugu Zhaiqi Suanfa《續古摘奇算法》 (Sequel to Excerpts of Mathematical Wonders) of 1275.

 

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.

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