{"id":86121,"date":"2023-04-29T20:14:24","date_gmt":"2023-04-29T17:14:24","guid":{"rendered":"https:\/\/azbuki.bg\/?p=86121"},"modified":"2025-07-08T10:02:20","modified_gmt":"2025-07-08T07:02:20","slug":"algorithms-for-construction-and-enumeration-of-closed-knights-paths","status":"publish","type":"post","link":"https:\/\/mathinfo.azbuki.bg\/en\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/","title":{"rendered":"Algorithms for Construction and Enumeration of Closed Knight\u2019s Paths"},"content":{"rendered":"<p><strong>Stoyan Kapralov1), Valentin Bakoev2), <\/strong><br \/>\n<strong>Kaloyan Kapralov3)<\/strong><br \/>\n<em>1)University of Gabrovo (Bulgaria)<\/em><br \/>\n<em>2)\u201cSt. Cyril and St. Methodius\u201d University of Veliko Tarnovo (Bulgaria)<\/em><br \/>\n<em>3)Sofia (Bulgaria)<\/em><\/p>\n<p><a href=\"https:\/\/doi.org\/10.53656\/math2023-2-1-alg\" target=\"_blank\" rel=\"noopener\">https:\/\/doi.org\/10.53656\/math2023-2-1-alg<\/a><\/p>\n<p><strong>Abstract.<\/strong> Two algorithms for constructing all closed knight\u2019s paths of lengths up to 16 are presented. An approach for classification (up to equivalence) of all such paths is considered. Two closed knight\u2019s paths are called equivalent if one can be obtained from the other by applying one or more of the equivalences: translation, rotation, symmetry, or when the corresponding polygons (whose vertices are the cells visited by the knight), are geometrically congruent. By applying the construction algorithms and classification approach, we enumerate both nonequivalent and non-self-intersecting knight\u2019s paths and show the obtained results. Some pedagogical aspects related to the problems under consideration and the teaching of subjects such as \u201cProgramming\u201d, \u201cAlgorithms and Data Structures\u201d, \u201cGraph Algorithms\u201d and \u201cCompetitive Programming\u201d are also discussed.<br \/>\n<em>Keywords:<\/em> knight graph; closed knight\u2019s path; nonequivalent path; non-self-intersecting path; equivalence; enumeration<\/p>\n<a href=\"https:\/\/mathinfo.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a>","protected":false},"excerpt":{"rendered":"<p>Stoyan Kapralov1), Valentin Bakoev2), Kaloyan Kapralov3) 1)University of Gabrovo (Bulgaria) 2)\u201cSt. Cyril and St. Methodius\u201d University of Veliko Tarnovo (Bulgaria) 3)Sofia (Bulgaria) https:\/\/doi.org\/10.53656\/math2023-2-1-alg Abstract. Two algorithms for constructing all closed knight\u2019s paths of lengths up to 16 are presented. An approach for classification (up to equivalence) of all such paths is considered. Two closed knight\u2019s [&hellip;]<\/p>","protected":false},"author":124332423426818,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jnews-multi-image_gallery":[],"jnews_single_post":{"source_name":"","source_url":"","via_name":"","via_url":"","override_template":"0","override":[{"template":"1","single_blog_custom":"","parallax":"1","fullscreen":"1","layout":"right-sidebar","sidebar":"default-sidebar","second_sidebar":"default-sidebar","sticky_sidebar":"1","share_position":"bottom","share_float_style":"share-monocrhome","show_share_counter":"1","show_view_counter":"1","show_featured":"0","show_post_meta":"1","show_post_author":"0","show_post_author_image":"1","show_post_date":"0","post_date_format":"default","post_date_format_custom":"Y\/m\/d","show_post_category":"1","show_post_reading_time":"0","post_reading_time_wpm":"300","show_zoom_button":"0","zoom_button_out_step":"2","zoom_button_in_step":"3","show_post_tag":"1","show_prev_next_post":"1","show_popup_post":"1","number_popup_post":"3","show_author_box":"0","show_post_related":"0","show_inline_post_related":"0"}],"override_image_size":"0","image_override":[{"single_post_thumbnail_size":"crop-500","single_post_gallery_size":"crop-500"}],"trending_post":"0","trending_post_position":"meta","trending_post_label":"Trending","sponsored_post":"0","sponsored_post_label":"Sponsored by","sponsored_post_name":"","sponsored_post_url":"","sponsored_post_logo_enable":"0","sponsored_post_logo":"","sponsored_post_desc":"","disable_ad":"0"},"jnews_primary_category":{"id":"","hide":""}},"categories":[1],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Algorithms for Construction and Enumeration of Closed Knight\u2019s Paths - \u0410\u0437-\u0431\u0443\u043a\u0438<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Algorithms for Construction and Enumeration of Closed Knight\u2019s Paths - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\"Stoyan Kapralov1), Valentin Bakoev2), Kaloyan Kapralov3) 1)University of Gabrovo (Bulgaria) 2)\u201cSt. Cyril and St. Methodius\u201d University of Veliko Tarnovo (Bulgaria) 3)Sofia (Bulgaria) https:\/\/doi.org\/10.53656\/math2023-2-1-alg Abstract. Two algorithms for constructing all closed knight\u2019s paths of lengths up to 16 are presented. An approach for classification (up to equivalence) of all such paths is considered. Two closed knight\u2019s [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/\" \/>\n<meta property=\"og:site_name\" content=\"\u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/Azbuki55\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-04-29T17:14:24+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-08T07:02:20+00:00\" \/>\n<meta name=\"author\" content=\"v.genkov@azbuki.bg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"v.genkov@azbuki.bg\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/\"},\"author\":{\"name\":\"v.genkov@azbuki.bg\",\"@id\":\"https:\/\/azbuki.bg\/#\/schema\/person\/92cc38d6a11fb032bf6299efd22a71c5\"},\"headline\":\"Algorithms for Construction and Enumeration of Closed Knight\u2019s Paths\",\"datePublished\":\"2023-04-29T17:14:24+00:00\",\"dateModified\":\"2025-07-08T07:02:20+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/\"},\"wordCount\":203,\"publisher\":{\"@id\":\"https:\/\/azbuki.bg\/#organization\"},\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/\",\"url\":\"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/\",\"name\":\"Algorithms for Construction and Enumeration of Closed Knight\u2019s Paths - 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Two algorithms for constructing all closed knight\u2019s paths of lengths up to 16 are presented. An approach for classification (up to equivalence) of all such paths is considered. Two closed knight\u2019s [&hellip;]","og_url":"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/","og_site_name":"\u0410\u0437-\u0431\u0443\u043a\u0438","article_publisher":"https:\/\/www.facebook.com\/Azbuki55\/","article_published_time":"2023-04-29T17:14:24+00:00","article_modified_time":"2025-07-08T07:02:20+00:00","author":"v.genkov@azbuki.bg","twitter_card":"summary_large_image","twitter_misc":{"Written by":"v.genkov@azbuki.bg","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/#article","isPartOf":{"@id":"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/"},"author":{"name":"v.genkov@azbuki.bg","@id":"https:\/\/azbuki.bg\/#\/schema\/person\/92cc38d6a11fb032bf6299efd22a71c5"},"headline":"Algorithms for Construction and Enumeration of Closed Knight\u2019s Paths","datePublished":"2023-04-29T17:14:24+00:00","dateModified":"2025-07-08T07:02:20+00:00","mainEntityOfPage":{"@id":"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/"},"wordCount":203,"publisher":{"@id":"https:\/\/azbuki.bg\/#organization"},"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/","url":"https:\/\/mathinfo.azbuki.bg\/uncategorized\/algorithms-for-construction-and-enumeration-of-closed-knights-paths\/","name":"Algorithms for Construction and Enumeration of Closed Knight\u2019s Paths - 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