![\"\"](\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2023\/11\/di-3-iamond-1.png\")
<\/a><\/figure><\/div>\n\n\n3-coloring the triamonds here isn’t hard. The tiling seems to want to have a bunch of points where four triamonds meet, which disrupts chains of forced colors. The challenge is adding more challenges on top of 3-coloring. I suspect that there is no strict 3-coloring of the triamonds. One possibility is a sort of meta-coloring of the di-triamonds where no two di-triamonds with the same color pair may be adjacent. The above diagram doesn’t qualify because there are blue-red di-triamonds touching each other. Problem #60<\/strong>: Find such a meta-coloring.<\/p>\n\n\n\n The diamonds in the tri-diamonds are even easier to 3-color. Enough so that 3-coloring them so each tri-diamond has all three colors (the equivalent of the poorly thought out problem #58 with the tri-dominoes) was no challenge at all. Perhaps there is something to be done with symmetry. Notice that, ignoring color and the tri-diamond outlines, the diamonds in the figure below have an an axis of reflection symmetry. I wonder if, for some tiling, some form of symmetry on the diamonds is possible where colors are included.<\/p>\n\n\n The meta-coloring idea above suggests a way to salvage Problem #58. Instead of a three coloring of the dominoes in a tri-domino tiling, we could look for a 4-coloring of the dominoes where every tri-domino contains 3 of the 4 colors, and there is simultaneously a meta-4-coloring of the tri-dominoes where no two adjacent tri-dominoes are missing the same color.<\/p>\n","protected":false},"excerpt":{"rendered":" Here’s a nice coincidence: the numbers of tri-diamonds and di-triamonds are both 9, which is the right amount to tile a regular hexagon of side length 3. And both sets can! Behold the di-triamonds: 3-coloring the triamonds here isn’t hard. The tiling seems to want to have a bunch of points where four triamonds meet, … Continue reading Component Colorings II: Diamonds and Triamonds<\/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":[9,249,248,17,218],"class_list":["post-1072","post","type-post","status-publish","format-standard","hentry","category-recreational-mathematics","tag-coloring","tag-component-coloring","tag-di-triamonds","tag-polyiamonds","tag-tri-diamonds"],"_links":{"self":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1072","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=1072"}],"version-history":[{"count":2,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1072\/revisions"}],"predecessor-version":[{"id":1178,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1072\/revisions\/1178"}],"wp:attachment":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/media?parent=1072"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/categories?post=1072"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/tags?post=1072"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}<\/a><\/figure><\/div>\n\n\n