{"id":1137,"date":"2024-04-01T22:59:26","date_gmt":"2024-04-02T05:59:26","guid":{"rendered":"https:\/\/puzzlezapper.com\/blog\/?p=1137"},"modified":"2024-04-01T22:59:26","modified_gmt":"2024-04-02T05:59:26","slug":"edgematching-to-the-stars","status":"publish","type":"post","link":"https:\/\/puzzlezapper.com\/blog\/2024\/04\/edgematching-to-the-stars\/","title":{"rendered":"Edgematching to the Stars"},"content":{"rendered":"\n

The best known combinatorially complete set of edgematching tiles are the 24 squares with 3 edge colors discovered by Percy MacMahon. These can be rotated, but not flipped over. I showed an illustration of them when I first discussed<\/a> edgematching puzzles on this blog. And now I’ll do it again!<\/p>\n\n\n

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Can we find an even more basic set? After all, 3\u00d73 square edgematching puzzles were common as advertising promotions in the first half of the 20th century. A set of 24 tiles seems excessive. And surely 2 edge colors would be simpler than 3.<\/p>\n\n\n\n

The problem is that with 2 colors, you get only 6 tiles, and they don’t make an interesting puzzle at all. But if you use tiles with fixed orientation, you get 16 tiles. Per David Singmaster’s Sources in Recreational Mathematics<\/a>, this set was described by J. J. M. Verbakel in 1975. We’ll use a line pointing out of a side to represent one of the colors, and a blank edge to represent the other, as this convention will help us to make better visual sense of some of the later figures.<\/p>\n\n\n

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They still don’t make a very good puzzle, as the above solution is basically trivial, so even adding the restriction that only one color can touch the border adds no challenge. But as a starting point for further exploration, they lead in some very interesting directions. Christian Freeling shows some of these on his page<\/a> on the “BackSlide” puzzle, which turns the pieces into a double-sided sliding 15-puzzle. (Freeling’s site also contains information about variations of these puzzles using hexagons<\/a> as well as squares with diagonal connections<\/a>.) <\/p>\n\n\n\n

The binary nature of the colors allows us break out of the 4\u00d74 square. Instead of both colors matching themselves, we can arrange the tiles into sets of polyominoes where one color matches itself, and the other matches an outer edge. Freeling calls these “transcendental” solutions. Finding these is still not much of a puzzle, but any unusual polyomino recreation is appreciated around here! <\/p>\n\n\n

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Breaking up the tiles further leads to the “starmaps” discovered (in the hexagonal context) by Martin Medema. Here, we place these tiles on an 8\u00d78 board. A line pointing out of a piece must connect, at some distance, to one pointing out of another piece. An empty edge must see only empty space in its direction. Conveniently, there are 32 empty edges, exactly as many as edges in the perimeter of the board. Freeling states that it is “considered good form” for no tiles to be adjacent. If we don’t use good form, it is not hard to transform a regular transcendental solution into a starmap by breaking it apart until no empty edges see each other. Finding a good form starmap isn’t a trivial puzzle at all.<\/p>\n\n\n

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My impetus for this post was an idea I had for a new type of starmap. We might require one edge type to match an adjacent instance of itself, and have an additional edge type that must see itself from more than one square away, enforcing “good form” for that type. The third edge type must see only empty space as before. There would be 81 fixed orientation pieces with three edge types, which is way more than I want to deal with. Surely there must be some better set with three edge types.<\/p>\n\n\n\n

Right, the MacMahon tiles! There are 32 instances of each edge type, so we can use an 8\u00d78 board again. I found this puzzle to be quite challenging.<\/p>\n\n\n

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Puzzles with sparse (or sparse-but-clumpy) piece placements are unusual and visually distinctive, and I hope to encounter more like these. <\/p>\n","protected":false},"excerpt":{"rendered":"

The best known combinatorially complete set of edgematching tiles are the 24 squares with 3 edge colors discovered by Percy MacMahon. These can be rotated, but not flipped over. I showed an illustration of them when I first discussed edgematching puzzles on this blog. And now I’ll do it again! Can we find an even … Continue reading Edgematching to the Stars<\/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":[186,185,260,261],"class_list":["post-1137","post","type-post","status-publish","format-standard","hentry","category-recreational-mathematics","tag-edgematching","tag-macmahon-squares","tag-starmaps","tag-transcendental-edgematching"],"_links":{"self":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1137","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=1137"}],"version-history":[{"count":5,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1137\/revisions"}],"predecessor-version":[{"id":1148,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1137\/revisions\/1148"}],"wp:attachment":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/media?parent=1137"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/categories?post=1137"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/tags?post=1137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}