AI在幻方中的应用
Applications of AI for Magic Squares
摘要：
在娱乐数学中，一个正规幻方是一个$n \times n$方阵，它的元素明显是整数$1 \ldots n^2$，这样每行、列、主迹和次迹都是一个常数$μ$。已证明有7，040个四阶正规幻方和2，202，441，792个五阶正规幻方，更高阶的尚未证实。以前的工作与四阶正规平方表明，对称性，如二面体群的存在，并在一定条件下，正常的幻方可以分为四个不同的子集。随着一个有效的回溯算法的实施以及监督机器学习技术的分类，它将被证明，整个四阶正常幻方可以通过扩展95个不对称的父母的对称群生成。讨论将表明，在这个项目中采用的方法可以同样适用于更高阶。
英文摘要：
In recreational mathematics, a normal magic square is an $n \times n$ square matrix whose entries are distinctly the integers $1 \ldots n^2$, such that each row, column, and major and minor traces sum to one constant $μ$. It has been proven that there are 7,040 fourth order normal magic squares and 2,202,441,792 fifth order normal magic squares, with higher orders unconfirmed. Previous work related to fourth order normal squares has shown that symmetries such as the dihedral group exist and that (under certain conditions) normal magic squares can be categorized into four distinct subsets. With the implementation of an efficient backtracking algorithm along with supervised machine learning techniques for classification, it will be shown that the entire set of fourth order normal magic squares can be generated by expanding the symmetry groups of 95 asymmetric parents. Discussion will suggest that methods employed in this project could similarly apply to higher orders.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
