论幻映射、幻群与幻方

Magic Mapping, Magic Group And Magic Square

给出幻映射、幻群及幻方与幻方同构的概念,把许多组合数学对象,如拉丁方、欧拉方及各种类型的幻方等等均纳入到幻映射这个统一的框架中。利用幻群对任何抽象幻方进行分类并证明任何一一幻方的解数均是与之有关的幻群的阶数的倍数。特别证明,Z16上的不同构4阶幻方只有220个。

This paper presents the concepts of abstract magic mapping (MM), magic group (MG), magic square (MS) and magic isomorphism. We take an abstract magic square as a graph of magic mapping and define a magic group as an especial magic mapping. The concept of magic mapping reflects essential of a great many of combination objects such as Latin squares, Eulerian squares, multiform magic squares and so on. We have proved some base theorems of abstract MM, MG and MS. Especially, we have proved that the total number of magic squares of the fourth order not to be isomorphic is 220.