Magic circle (mathematics)

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Yang Hui's Magic Circle
Yang Hui's Magic Circle

Magic circles were invented by Southern Song dynasty mathematician Yang Hui:one of his magic circles was constructed from 33 natural numbers from 1 to 33 arranged on four circles , with 9 at the center.

[edit] Yang Hui magic circles

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 《续古摘奇算法》 (Sequel to Excerpts of Mathematical Wonders).

[edit] Ding Yidong magic circles

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.



[edit] References

  • 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
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