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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.20017 |
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| _version_ | 1866918272523304960 |
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| author | Pamplona, João Vitor Pinheiro, Maria Eduarda Santos, Luiz-Rafael |
| author_facet | Pamplona, João Vitor Pinheiro, Maria Eduarda Santos, Luiz-Rafael |
| contents | Magic squares are a fascinating mathematical challenge that has intrigued mathematicians for centuries. Given a positive (and possibly large) integer \( n \), one of the main challenges that still remains is to find, within a computational time, a magic square of order \( n \), that is, a square matrix of order \( n \) with unique integers from \( a_{\min} \) to \( a_{\max} \), such that the sum of each row, column, and diagonal equals a constant \( \mathcal{C}(A) \). In this work, we first present an integer constraint satisfaction problem for constructing a magic square of order \( n \). Nonetheless, the solution time of this problem grows exponentially as the order increases. To overcome this limitation, we also propose a that constructs magic squares depending on whether \( n \) is odd, singly even, or doubly even. Moreover, we provide a proof of the correctness of this novel approach. Our numerical results show that our method can construct magic squares of order up to \num{70000} in less than \num{140} seconds, demonstrating its efficiency and scalability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20017 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructing Magic Squares: an integer constraint satisfaction problem and a fast approach Pamplona, João Vitor Pinheiro, Maria Eduarda Santos, Luiz-Rafael Optimization and Control Magic squares are a fascinating mathematical challenge that has intrigued mathematicians for centuries. Given a positive (and possibly large) integer \( n \), one of the main challenges that still remains is to find, within a computational time, a magic square of order \( n \), that is, a square matrix of order \( n \) with unique integers from \( a_{\min} \) to \( a_{\max} \), such that the sum of each row, column, and diagonal equals a constant \( \mathcal{C}(A) \). In this work, we first present an integer constraint satisfaction problem for constructing a magic square of order \( n \). Nonetheless, the solution time of this problem grows exponentially as the order increases. To overcome this limitation, we also propose a that constructs magic squares depending on whether \( n \) is odd, singly even, or doubly even. Moreover, we provide a proof of the correctness of this novel approach. Our numerical results show that our method can construct magic squares of order up to \num{70000} in less than \num{140} seconds, demonstrating its efficiency and scalability. |
| title | Constructing Magic Squares: an integer constraint satisfaction problem and a fast approach |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2504.20017 |