An Enhanced Multi-Stage Evolution Strategy for Large Size Magic Square Generationand Extension to Magic Rectangles

An Enhanced Multi-Stage Evolution Strategy for Large Size Magic Square Generationand Extension to Magic Rectangles | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article An Enhanced Multi-Stage Evolution Strategy for Large Size Magic Square Generationand Extension to Magic Rectangles Kazuki Takemi, Takuto Sakuma, Shohei Kato This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7917518/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 02 Apr, 2026 Read the published version in Evolutionary Intelligence → Version 1 posted 9 You are reading this latest preprint version Abstract This research presents an enhanced multi-stage evolutionary strategy for the rapid and stable generation of large-scale magic squares and rectangles. Although previous methods have demonstrated improved performance, significant challenges remain in their scalability and scope of application. Our proposed method addresses these limitations through three key contributions. First, we address a key bottleneck in odd-order square generation by introducing a deterministic initialization method that integrates a "3-column block" strategy with the "adjacent-pair uniformity method." This approach reduces the initial execution time to tens of milliseconds, rendering it nearly independent of the square’s size , a dramatic improvement over previous methods, where time increased linearly with the order. Second, we substantially advanced the search algorithm by enhancing the mutation operator to allow swapping of up to two cell pairs and dynamically adjusts its strategy based on the search progress. Furthermore, incorporating a differential update into the fitness evaluation reduces computational complexity to approximately O(1), eliminating costly full recalculations. These synergistic improvements reduce the required generations by approximately 40% for large-scale problems and shorten the execution time by a factor of several hundred times compared with conventional methods. Third, the algorithm was extended to generate magic rectangles, an advancement beyond prior work limited to square matrices. This versatility enables practical applications in fields, such as image encryption, that involve non-square data. In summary, our method significantly enhances the performance of magic square generation, thereby providing a practical and scalable foundation for future research. Magic Square Semi-magic Square Evolutionary Strategy and Multi-stage Evolutionary strategy and magic rectangles Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 02 Apr, 2026 Read the published version in Evolutionary Intelligence → Version 1 posted Editorial decision: Revision requested 09 Dec, 2025 Reviews received at journal 05 Dec, 2025 Reviewers agreed at journal 01 Dec, 2025 Reviews received at journal 22 Nov, 2025 Reviewers agreed at journal 17 Nov, 2025 Reviewers invited by journal 17 Nov, 2025 Editor assigned by journal 22 Oct, 2025 Submission checks completed at journal 22 Oct, 2025 First submitted to journal 21 Oct, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7917518","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":549196024,"identity":"cad8a8ec-85b4-40fa-bf4c-8c7827072143","order_by":0,"name":"Kazuki Takemi","email":"","orcid":"","institution":"Nagoya Institute of Technology","correspondingAuthor":false,"prefix":"","firstName":"Kazuki","middleName":"","lastName":"Takemi","suffix":""},{"id":549196025,"identity":"300c5a17-5d2a-4b4b-a46d-c65d07ede232","order_by":1,"name":"Takuto Sakuma","email":"","orcid":"","institution":"Nagoya Institute of Technology","correspondingAuthor":false,"prefix":"","firstName":"Takuto","middleName":"","lastName":"Sakuma","suffix":""},{"id":549196026,"identity":"ebf1ce56-5090-443d-9340-6cd94808c4ac","order_by":2,"name":"Shohei Kato","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCElEQVRIie2QsWrDMBCGTwjUxdSrjEF9hSsaQt5GwnsJZCm0UBeDuvgBDHmJPEKCIVkKWQMJ1CbQOdAlUA+9tjQQqAzZStE3SJy4j/9OAIHAH0R+XzNAYI8NnT9PokfhR6VAQDxLAUHNR8VP8rSaN9Bt1eCicHejUafiSc72B7i88SlplHFk7lUPy7nbVIhabmc8KUGMfYqCTEiW13a6tm4TIdqpNJDSlDb3KfGOVuhIeWndmJQHUvh7n5JKSgHxmcIcJ8WgNKI3Jal2Gq2rNT7bIo1QX1c04bBE/y5yZdtm39UKl8v2LerUVVxl9fpwu/D+2BfmtGQ0Ei7ML4393J+vBAKBwH/lA9eKUtl0QdpmAAAAAElFTkSuQmCC","orcid":"","institution":"Nagoya Institute of Technology","correspondingAuthor":true,"prefix":"","firstName":"Shohei","middleName":"","lastName":"Kato","suffix":""}],"badges":[],"createdAt":"2025-10-22 08:10:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7917518/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7917518/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s12065-026-01171-w","type":"published","date":"2026-04-02T16:00:12+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":96777513,"identity":"c93b2cce-38cf-4041-a857-2ecd12bd0928","added_by":"auto","created_at":"2025-11-26 03:25:01","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5640,"visible":true,"origin":"","legend":"","description":"","filename":"08e6fb8e2a36456a865e3f2257c56450.json","url":"https://assets-eu.researchsquare.com/files/rs-7917518/v1/50f74ae99ee887a4bd4100f7.json"},{"id":106344800,"identity":"ef129f23-5f35-4320-b658-054e3ba1c5c1","added_by":"auto","created_at":"2026-04-07 16:16:54","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":268240,"visible":true,"origin":"","legend":"","description":"","filename":"EItexsource2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7917518/v1_covered_d9203073-55d2-496c-a1c0-c1d892944d53.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"An Enhanced Multi-Stage Evolution Strategy for Large Size Magic Square Generationand Extension to Magic Rectangles","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"evolutionary-intelligence","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"evin","sideBox":"Learn more about [Evolutionary Intelligence](http://link.springer.com/journal/12065)","snPcode":"12065","submissionUrl":"https://submission.nature.com/new-submission/12065/3","title":"Evolutionary Intelligence","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Magic Square, Semi-magic Square, Evolutionary Strategy,and Multi-stage Evolutionary strategy, and magic rectangles","lastPublishedDoi":"10.21203/rs.3.rs-7917518/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7917518/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This research presents an enhanced multi-stage evolutionary strategy for the rapid and stable generation of large-scale magic squares and rectangles. 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