Scholarlink

<p>This paper proposes a multi-stage evolutionary approach for generating large-scale magic squares. We introduce three main contributions: (1) a deterministic initialization reducing Stage-1 time to tens of milliseconds across all tested matrix sizes; (2) an improved search algorithm that reduces the computational complexity of each fitness evaluation to <i>O</i>(1) through differential updates and dynamic mutation strategies, achieving approximately 40% fewer generations and speedups exceeding 100-fold for large instances; and (3) an extension to arbitrary magic rectangles, broadening the applicability of the proposed framework to practical domains such as image encryption. Experiments on magic squares up to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(400 \times 400\)</EquationSource> </InlineEquation> and magic rectangles up to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(100 \times 300\)</EquationSource> </InlineEquation> demonstrate practical scalability and efficiency.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

An enhanced multi-stage evolution strategy for large size magic square generation and extension to magic rectangles

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年4月2日

查看全文
  • Kazuki Takemi,
  • Takuto Sakuma,
  • Shohei Kato

摘要

This paper proposes a multi-stage evolutionary approach for generating large-scale magic squares. We introduce three main contributions: (1) a deterministic initialization reducing Stage-1 time to tens of milliseconds across all tested matrix sizes; (2) an improved search algorithm that reduces the computational complexity of each fitness evaluation to O(1) through differential updates and dynamic mutation strategies, achieving approximately 40% fewer generations and speedups exceeding 100-fold for large instances; and (3) an extension to arbitrary magic rectangles, broadening the applicability of the proposed framework to practical domains such as image encryption. Experiments on magic squares up to \(400 \times 400\) and magic rectangles up to \(100 \times 300\) demonstrate practical scalability and efficiency.