Yang Hui

Little Facts

I was born aroung 1238 and died around 1298. I am a Chinese mathematician from Qiantang and i wrote several mathematical texts. These contained solutions of quadratic equations as well as Pascal's triangle, magic squares, magic circles, and binomial theorem, and I am known for my contribution of presenting 'Yang Hui's Triangles'. I also served as an official under the southern Sung(The Sung Dynasty).

Magic Squares

In recreational mathematics, a magic square of order n is an arrangement of n² numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant.[1] A normal magic square contains the integers from 1 to n². The term "magic square" is also sometimes used to refer to any of various types of word square.Normal magic squares exist for all orders n ≥ 1 except n = 2, although the case n = 1 is trivial—it consists of a single cell containing the number 1. The smallest nontrivial case, shown below, is of order 3.

Magic Circles

One of my magic circles was constructed from 33 natural numbers from 1 to 33 arranged on four circles , with 9 at the center.

The magic circle has the following properties
The sum of the numbes on four diamters = 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.