Efficient generation of magic square using multi-stage evolutionary strategy cascaded by satisfiable individuals for magic sum

Abstract

The magic square has been pursued as a theme of recreational mathematics since ancient times. Most of existing methods are rule-based methods, so only a few special magic squares can be generated. This paper proposes a multi-stage evolutionary strategy that hierarchically classifies the constraints of magic squares into three stages of rows, columns, and diagonals, and uses the individuals satisfying each layer's constraints as the initial individuals for the next layer. A magic square is a state in which the sum of each element in a square matrix in rows, columns, and diagonals is equal to a constant value. A square matrix that satisfies only the constraints in rows and columns is called a semi-magic square. In this research, the semi-magic square was divided into two layers, and by adding a lower layer that generates individuals satisfying only the row constraints, the need for exploration in the row direction was eliminated, resulting in a significant reduction in the number of searches. Experiments compared the execution time and generation number between the proposed method and previous study, and achieved a significant reduction of the execution time.