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。

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

《周易》六十四卦的卦序问题至今尚无定论。运用幻方原理，结合宋人伏羲六十四卦方图，构造出六十四卦幻方图，建构了宇宙对称原理模型，并对这一模型的基本结构和原理进行描述与解释，对一阴一阳卦、二阴二阳卦、三阴三阳卦、六阴六阳卦、十二辟卦以及十六米字卦等排列与分布进行分析，得出“幻方原理决定古经卦序”的研究结论。最后，运用这一研究结论，尝试构造六十四卦马步幻方图，并进行验证。

Trigram sequence of the“Zhou Yi”sixty-four trigrams is inconclusive so far. This study uses magic square theory, and combined with the Fu Xi sixty-four trigrams square diagram, constructs the sixty-four trigrams magic map, and the universe symmetry principle model. The basic structure and principle of this model is described and explained, and the arrangement and distribution of one Yin and one Yang trigrams, two Yin and two Yang trigrams, three Yin and three Yang trigrams, six Yin and six Yang trigrams, twelve Pi trigrams and sixteen Mi-zi trigrams are analyzed, and reached the research conclusion of “magic principle decided the ancient trigram sequence”. At last, by using the conclusions of this research, the sixty-four trigrams horse step magic figure as verification is constructed.

研究了奇数阶幻方构造算法及其数据依赖性。通过对现有算法内存访问顺序和相关性进行分析，找出了限制现有算法并行性的主要因素，并通过改变内存访问模式和算法重构消除了数据依赖性。新的并行算法可以将原有算法的时间复杂度降为 O（n），并且该算法可以达到成本最优。给出了该算法在不同存储系统下的实现方法。

The algorithm and data dependency of original construction method for odd-order magic square are studied.By analyzed the memory access pattern and data dependency, and figured out the major limiter for construction of parallel algorithm, the memory access order and redesigned algorithm are designed to remove data dependency.The new algorithm decreases the time complex to O(n),and gets cost efficiency.Meanwhile,the implement methods are illustrated under different storage systems.