{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:35:57Z","timestamp":1760060157297,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,4]],"date-time":"2025-08-04T00:00:00Z","timestamp":1754265600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>There has been much interest in the mathematical investigation of critical sets and unavoidable sets in Latin Squares, Sudoku, and their applications to practical problems in areas such as agriculture and cryptology. This paper considers the associated structures of Strictly Concentric Magic Squares (SCMSs) and Prime Strictly Concentric Magic Squares (PSCMSs). A framework of formal definitions is given that leads to the definitions of critical sets and unavoidable sets. Minimal critical sets are of interest in Latin Squares, and in this article, the cardinality of minimal critical sets of SCMS is given for all n, n odd. Two families of unavoidable sets are established for SCMS, leading to a complete classification of unavoidable sets of minimum PSCMS of order 5.<\/jats:p>","DOI":"10.3390\/axioms14080607","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T08:46:55Z","timestamp":1754383615000},"page":"607","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Critical Sets and Unavoidable Sets of Strictly Concentric Magic Squares of Odd Order and Their Application to Prime Strictly Concentric Magic Squares of Order 5"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4364-1786","authenticated-orcid":false,"given":"Anna Louise","family":"Skelt","sequence":"first","affiliation":[{"name":"School of Computing and Mathematics, University of South Wales, Llantwit Rd, Pontypridd CF37 1DL, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6292-8731","authenticated-orcid":false,"given":"Stephanie","family":"Perkins","sequence":"additional","affiliation":[{"name":"School of Computing and Mathematics, University of South Wales, Llantwit Rd, Pontypridd CF37 1DL, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8695-9778","authenticated-orcid":false,"given":"Paul Alun","family":"Roach","sequence":"additional","affiliation":[{"name":"School of Computing and Mathematics, University of South Wales, Llantwit Rd, Pontypridd CF37 1DL, UK"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Johnson, L., and Perkins, S. 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Proceedings in Adaptation, Learning and Optimization, Proceedings of the International Conference on Intelligent Vision and Computing (ICIVC 2021), Online, 3\u20134 October 2021, Springer.","DOI":"10.1007\/978-3-030-97196-0"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/607\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:22:49Z","timestamp":1760034169000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/607"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,4]]},"references-count":20,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080607"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080607","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,8,4]]}}}