三阶幻方问题的代数解法

Algebra approach to solve third-order magic square question

介绍了中国古代数学家甄鸾和杨辉关于三阶幻方的一个构造方法。用线性代数的方法构造三阶幻方约束方程组时,当自由未知量分别取1,2,3,…,9时,对应的方程组共有56组整数解,为了找出幻方的全部解,需要确定自由未知量取值的限制条件,由此对方程组中的自由未知量的取值进行了分类探讨,进而求出三阶幻方的全部解。利用线性代数的方法去探求三阶幻方的解法种数显然不是最简捷的,但却有完备的理论做保证。此外,提供了一个如何利用线性代数知识来解决实际问题的实例,并简介了三阶幻方所具有的奇妙的特性。

This paper introduces a constructor of third - order magic square put forward by the two ancient Chinese mathematicians Yang Hui and Zhen Luan. 56 groups of integer solutions correspond the equations in total, when the third - order magic square constraint equations are constructed in a linear algebra way and the freedom unknowns are taken by 1, 2, 3,...., 9 respectively. To find all solutions in the magic square, restrictions of the freedom unknown values are to be ascertained. Thus, the values of the freedom unknown in the equations are discussed in classifications to calculate all the solutions of third - order magic square. The method of exploring the solutions of third - order magic square by using linear algebra methods is obvi- ously not the simplest, however, it can be guaranteed by a complete theory. In addition, the article pro- vides an example of how to use linear algebra to solve practical problems and briefly introduces the wonder- ful properties of the third - order magic square..