{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:13:05Z","timestamp":1760177585058,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,7,25]],"date-time":"2020-07-25T00:00:00Z","timestamp":1595635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002873","name":"Chulalongkorn University","doi-asserted-by":"publisher","award":["Research Assistantship Fund, Faculty of Science"],"award-info":[{"award-number":["Research Assistantship Fund, Faculty of Science"]}],"id":[{"id":"10.13039\/501100002873","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A (legal) knight\u2019s move is the result of moving the knight two squares horizontally or vertically on the board and then turning and moving one square in the perpendicular direction. A closed knight\u2019s tour is a knight\u2019s move that visits every square on a given chessboard exactly once and returns to its start square. A closed knight\u2019s tour and its variations are studied widely over the rectangular chessboard or a three-dimensional rectangular box. For m,n&gt;2r, an (m,n,r)-ringboard or (m,n,r)-annulus-board is defined to be an m\u00d7n chessboard with the middle part missing and the rim contains r rows and r columns. In this paper, we obtain that a (m,n,r)-ringboard with m,n\u22653 and m,n&gt;2r has a closed knight\u2019s tour if and only if (a) m=n=3 and r=1 or (b) m,n\u22657 and r\u22653. If a closed knight\u2019s tour on an (m,n,r)-ringboard exists, then it has symmetries along two diagonals.<\/jats:p>","DOI":"10.3390\/sym12081217","type":"journal-article","created":{"date-parts":[[2020,7,27]],"date-time":"2020-07-27T09:24:49Z","timestamp":1595841889000},"page":"1217","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Closed Knight\u2019s Tours on (m,n,r)-Ringboards"],"prefix":"10.3390","volume":"12","author":[{"given":"Wasupol","family":"Srichote","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0189-7799","authenticated-orcid":false,"given":"Ratinan","family":"Boonklurb","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sirirat","family":"Singhun","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ramkhamhaeng University, Bangkok 10240, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1080\/0025570X.2002.11953127","article-title":"Pillow chess","volume":"75","author":"Cairns","year":"2002","journal-title":"Math. 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