{"id":1107,"date":"2019-06-24T08:47:04","date_gmt":"2019-06-24T12:47:04","guid":{"rendered":"http:\/\/josmfs.net\/?p=1107"},"modified":"2024-04-10T17:47:35","modified_gmt":"2024-04-10T21:47:35","slug":"star-polygon-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2019\/06\/24\/star-polygon-problem\/","title":{"rendered":"Star Polygon Problem"},"content":{"rendered":"<p>This is problem #25 from the UKMT 2014 Senior Challenge.<\/p>\n<p>\u201cFigure 1 shows a tile in the form of a trapezium [trapezoid], where <em>a<\/em> = 83\u2153\u00b0. Several copies of the tile placed together form a symmetrical pattern, part of which is shown in Figure 2. The outer border of the complete pattern is a regular \u2018star polygon\u2019. Figure 3 shows an example of a regular \u2018star polygon\u2019.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-1105 aligncenter\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/06\/Star-Polygon-Prob-Fig.jpg\" alt=\"\" width=\"475\" height=\"130\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/06\/Star-Polygon-Prob-Fig.jpg 604w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/06\/Star-Polygon-Prob-Fig-300x82.jpg 300w\" sizes=\"auto, (max-width: 475px) 100vw, 475px\" \/>How many tiles are there in the complete pattern?<br \/>\n<span style=\"color: #ffffff;\">_____<\/span>A<span style=\"color: #ffffff;\">_<\/span>48<span style=\"color: #ffffff;\">_____<\/span>B<span style=\"color: #ffffff;\">_<\/span>54<span style=\"color: #ffffff;\">_____<\/span>C<span style=\"color: #ffffff;\">_<\/span>60<span style=\"color: #ffffff;\">_____<\/span>D<span style=\"color: #ffffff;\">_<\/span>66<span style=\"color: #ffffff;\">_____<\/span>E<span style=\"color: #ffffff;\">_<\/span>72\u201d<\/p>\n<p><a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/04\/Star-Polygon-Answer.jpg\">Answer<\/a>.<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2019\/06\/Star-Polygon-Problem-190622.pdf\">Star Polygon Problem<\/a> for solutions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is problem #25 from the UKMT 2014 Senior Challenge. \u201cFigure 1 shows a tile in the form of a trapezium [trapezoid], where a = 83\u2153\u00b0. Several copies of the tile placed together form a symmetrical pattern, part of which is shown in Figure 2. The outer border of the complete pattern is a regular [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[13,83],"class_list":["post-1107","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-plane-geometry","tag-ukmt-challenge"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1107","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=1107"}],"version-history":[{"count":6,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1107\/revisions"}],"predecessor-version":[{"id":3821,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/1107\/revisions\/3821"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=1107"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=1107"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=1107"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}