The construction of magic squares is an ancient mathematical problem with numerous applications in areas such as information transmission in network security. While there have been various approaches to constructing high-order magic squares, they are typically computationally intensive. We explore and adapt approaches with roots in Chinese classics to develop a set of simple yet comprehensive methods for constructing magic squares of any order. For generating odd-order magic squares, the traditional "Siamese method" (The Siamese Method(or De la Loubère Method): starting from the central box of the first row with the number 1 (or the first number of any arithmetic progression), the fundamental movement for filling the boxes is diagonally up and right (↗), one step at a time. When a move would leave the square, it is wrapped around to the last row or first column, respectively. If a filled box is encountered, one moves vertically down one box (↓) instead, then continuing as before.) was introduced over 300 years ago by Simon de la Loubère, a French ambassador to Thailand in later seventeenth century. While workable, the Siamese method is cumbersome and lacks an underlying logic (Benjamin et al. in Coll Math J 45:92–100, 2014). We employ a concept derived from the circular structure of Five Elements (The Five Elements (Wuxing) is one of the two fundamental Theories define Chinese civilization. It is characterized essentially by five components to interpret everything. In endless cycles, Five Elements are mutually generative, and at the same time mutually restrictive. For the generation relationship, Metal generates Water generates Wood generates Fire generates Earth generates Metal generates Water generates Wood……(see Fig. 1B below); For the restraint relationship, Metal destructs Wood destructs Earth destructs Water destructs Fire destructs Metal destructs Wood destructs Earth generates Water …… This key concept of dynamic and constant balancing, unbalancing, rebalancing process applies universally to explaining natural phenomena, cosmology, medicine, politics, and human affairs.) in Chinese culture to establish a prototype for constructing odd-order magic squares. This approach significantly simplifies the algorithm, and reveals the underlying mathematical logic and mechanism of the Siamese method. We also adapt the composition of the Bagua (The Yin-Yang is another fundamental theory of Chinese thoughts, originating from Yijing. It is characterized with the Yin-Yang duality and interdependence of two for all things in the universe. Where Yin represents the passive, dark, cold, feminine, and receptive aspects of nature; Yang represents the active, bright, hot, masculine, and expansive aspects of nature. Yin and Yang are contrast yet complementary. The Yin Sphere and Yang Sphere are in opposition, eternally embracing together, moving toward and transforming into each other [<CitationRef CitationID="CR6">6</CitationRef>].) (eight trigrams in Yijing) from the ancient wisdom in Chinese classics to develop a prototype for constructing magic squares of 2 k order as reported (Zongxi in Yixue Xiangshulun, Jiuzhou Publishing House, 2007; Zhu in A history of Yi philosophy, Peking University Press, 1986). This in turn enables us to establish a comprehensive algorithm to cover the construction of magic squares of all orders. Our comprehensive algorithm is capable of constructing extremely high-order magic squares, which are of a high degree of symmetricity and ready for generating countless variations (McCranie in Math Teach 81:674–678, 1988). It therefore holds promise to be applied in network security for data transmission, and other applications, such as password encryption, etc.

New Algorithms for Constructing Magic Squares with Ancient Chinese Wisdoms

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发表日期：

2025年5月16日

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Dee Bruce Sun,
Stephen Sun

摘要

The construction of magic squares is an ancient mathematical problem with numerous applications in areas such as information transmission in network security. While there have been various approaches to constructing high-order magic squares, they are typically computationally intensive. We explore and adapt approaches with roots in Chinese classics to develop a set of simple yet comprehensive methods for constructing magic squares of any order. For generating odd-order magic squares, the traditional "Siamese method" (The Siamese Method(or De la Loubère Method): starting from the central box of the first row with the number 1 (or the first number of any arithmetic progression), the fundamental movement for filling the boxes is diagonally up and right (↗), one step at a time. When a move would leave the square, it is wrapped around to the last row or first column, respectively. If a filled box is encountered, one moves vertically down one box (↓) instead, then continuing as before.) was introduced over 300 years ago by Simon de la Loubère, a French ambassador to Thailand in later seventeenth century. While workable, the Siamese method is cumbersome and lacks an underlying logic (Benjamin et al. in Coll Math J 45:92–100, 2014). We employ a concept derived from the circular structure of Five Elements (The Five Elements (Wuxing) is one of the two fundamental Theories define Chinese civilization. It is characterized essentially by five components to interpret everything. In endless cycles, Five Elements are mutually generative, and at the same time mutually restrictive. For the generation relationship, Metal generates Water generates Wood generates Fire generates Earth generates Metal generates Water generates Wood……(see Fig. 1B below); For the restraint relationship, Metal destructs Wood destructs Earth destructs Water destructs Fire destructs Metal destructs Wood destructs Earth generates Water …… This key concept of dynamic and constant balancing, unbalancing, rebalancing process applies universally to explaining natural phenomena, cosmology, medicine, politics, and human affairs.) in Chinese culture to establish a prototype for constructing odd-order magic squares. This approach significantly simplifies the algorithm, and reveals the underlying mathematical logic and mechanism of the Siamese method. We also adapt the composition of the Bagua (The Yin-Yang is another fundamental theory of Chinese thoughts, originating from Yijing. It is characterized with the Yin-Yang duality and interdependence of two for all things in the universe. Where Yin represents the passive, dark, cold, feminine, and receptive aspects of nature; Yang represents the active, bright, hot, masculine, and expansive aspects of nature. Yin and Yang are contrast yet complementary. The Yin Sphere and Yang Sphere are in opposition, eternally embracing together, moving toward and transforming into each other [6].) (eight trigrams in Yijing) from the ancient wisdom in Chinese classics to develop a prototype for constructing magic squares of 2^k order as reported (Zongxi in Yixue Xiangshulun, Jiuzhou Publishing House, 2007; Zhu in A history of Yi philosophy, Peking University Press, 1986). This in turn enables us to establish a comprehensive algorithm to cover the construction of magic squares of all orders. Our comprehensive algorithm is capable of constructing extremely high-order magic squares, which are of a high degree of symmetricity and ready for generating countless variations (McCranie in Math Teach 81:674–678, 1988). It therefore holds promise to be applied in network security for data transmission, and other applications, such as password encryption, etc.