用代数方法探讨四阶幻方的解

从杨辉四阶幻方入手,介绍了两种四阶幻方的构造方法,分别是通过对幻方进行元素互换的杨辉构造法和用元素构造矩阵的矩阵构造法。运用线性代数的方法探求四阶幻方的解,建立了四阶幻方的约束方程组,并通过初等变换得到了约束方程组等价的约束条件,利用这些约束条件并结合四阶幻方的性质得到了关于四阶幻方的等价关系。通过这种等价关系,对四阶幻方进行＂行变换＂与＂列变换＂举出了由已知幻方生成基本幻方和怎样构造四阶幻方的例子。阐述了幻方同构的概念和幻方总数与基本解的个数,并且指出对于一个已知幻方,共存在8个与其同构的幻方,其中包括已知幻方。

Solution of four order magic square with algebraic method

The paper begins to the Yang Hui fourth-order magic square and introduces two methods of constructing magic square of fourth-order,they are the Yang Hui method though exchanging the elements of magic and the constructing matrix with elements.By using the method of linear algebra to explore the solution of fourth-order magic squares and establishing a constraint equations of fourth-order magic square to obtain the equivalent constraints of equations though elementary transformation.The paper obtains the equivalence relation of four order by using these constraints and the properties of magic squares.Though the above equivalence relation,we made the forth-order magic squares to＂line transformation＂and＂row transformation＂and cited the examples to illustrate that how to use the known to get the basic magic square.Finally,the paper described the isomorphic concept of magic square and demonstrated the number of basic solution.As to a known magic square,the paper points that there are eight isomorphic magic squares which including the known magic square.