{"id":603,"date":"2017-02-03T17:15:05","date_gmt":"2017-02-04T01:15:05","guid":{"rendered":"http:\/\/puzzlezapper.com\/blog\/?p=603"},"modified":"2017-02-03T17:15:05","modified_gmt":"2017-02-04T01:15:05","slug":"pentominoes-on-paths-and-trees","status":"publish","type":"post","link":"https:\/\/puzzlezapper.com\/blog\/2017\/02\/pentominoes-on-paths-and-trees\/","title":{"rendered":"Pentominoes on paths and trees"},"content":{"rendered":"<p>Here&#8217;s a path that could be taken by a chess king. All subpaths of length four describe a different pentomino:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2017\/02\/5omino-mopping.gif\" alt=\"A pentomino train\" width=\"202\" height=\"202\" class=\"aligncenter size-full wp-image-604\" \/><\/p>\n<p>This led from the grid of pentomino painting instructions that I posted previously. Consider a string of arrows for which the subsequences of length 4 include instructions for producing all 12 pentominoes. (This is somewhat analogous to a de Bruijn sequence.) For the case shown, the string is \u2190\u2191\u2196\u2192\u2192\u2192\u2192\u2193\u2193\u2190\u2193\u2190\u2198\u2197\u2193, although the graphic seems more illuminating than the arrow string here.<\/p>\n<p>Instead of a path, we could have a tree of pentominoes:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2017\/02\/5omino-tree-doodle-1.gif\" alt=\"\" width=\"321\" height=\"241\" class=\"aligncenter size-full wp-image-609\" \/><\/p>\n<p>Along with the constraints that each pentomino occurs exactly once, and no square is used more than once, I wanted to limit the number of branches per node. The root node having three branches might be considered a flaw, but this was the best I could do.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here&#8217;s a path that could be taken by a chess king. All subpaths of length four describe a different pentomino: This led from the grid of pentomino painting instructions that I posted previously. Consider a string of arrows for which the subsequences of length 4 include instructions for producing all 12 pentominoes. (This is somewhat &hellip; <a href=\"https:\/\/puzzlezapper.com\/blog\/2017\/02\/pentominoes-on-paths-and-trees\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Pentominoes on paths and trees<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[101,10,39,180],"class_list":["post-603","post","type-post","status-publish","format-standard","hentry","category-recreational-mathematics","tag-de-bruijn-sequences","tag-pentominoes","tag-polyform-change-paths","tag-trees"],"_links":{"self":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/603","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/comments?post=603"}],"version-history":[{"count":6,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/603\/revisions"}],"predecessor-version":[{"id":612,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/603\/revisions\/612"}],"wp:attachment":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/media?parent=603"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/categories?post=603"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/tags?post=603"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}