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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 81217, 1431]*) (*NotebookOutlinePosition[ 81894, 1454]*) (* CellTagsIndexPosition[ 81850, 1450]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Of some new acyclic canons", "Title"], Cell["Talk at MaMuX seminar Ircam, Paris, january the 24th", "Subtitle"], Cell[TextData[{ "The new canons where obtained by some of the following actions :\n\ \[Bullet] completion (given R, find S)\n\[Bullet] affine orbite (use the \ affine maps ", Cell[BoxData[ \(TraditionalForm\`x \[Rule] k\[Times]x\ mod\ n\)]], " to get other tiles from a given one)\n\[Bullet] \ augmentations/diminutions. On a tile, it works in one of two ways :\n\t\:fdff \ Replace each note (or silence) by ", Cell[BoxData[ \(TraditionalForm\`p\)]], " consecutive notes (or silences)\n\t\:fdff Change the tempo by the factor \ ", Cell[BoxData[ \(TraditionalForm\`p\)]], "." }], "Section"], Cell["\<\ Solutions are given in intervalic form, reduced up to cyclic \ permutation.\ \>", "Section"], Cell[TextData[{ "The relevant functions are written with ", StyleBox["Mathematica", FontSlant->"Italic"], " in canonCrawler.nb. There names are respectively :\n\[Bullet] ", StyleBox["complete", "Input"], " (or randomCanonFromList, for the quicker but randomized version)\n\ \[Bullet] ", StyleBox["orbite", "Input"], "\n\[Bullet] ", StyleBox["zoomIn, zoomOut", "Input"], " (and obviously ", Cell[BoxData[ FormBox[ StyleBox[\(tile -> p\[Times]tile\), "Input"], TraditionalForm]]], ")\n\[Bullet] ", StyleBox["basicForm", "Input"], " (i.e. first element of orbite)" }], "Section"], Cell[CellGroupData[{ Cell["\<\ In the following, a few words are given to explain the howTo. The \ solutions are listed next.\ \>", "Section"], Cell[CellGroupData[{ Cell["144=6\[Times]24 :", "Subsection"], Cell[TextData[{ "\[Bullet]I cannot complete a 6 notes motive on period 144. The reverse, \ using any element (24 long) of ", Cell[BoxData[ \(TraditionalForm\`R\_\(144 bis\)\)]], ", gives back only the 3 known elements of ", Cell[BoxData[ \(TraditionalForm\`S\_\(144 bis\)\)]], ".\n\[Bullet] Something weird : doubling ", Cell[BoxData[ \(TraditionalForm\`S\_72\)]], " gives ", Cell[BoxData[ \(TraditionalForm\`\(\(2 S\_72\)\(\ \)\(=\)\(\ \)\)\)]], Cell[BoxData[ \(TraditionalForm\`S\_\(144 bis\)\)]], "already given by Vuza's algorithm: nothing new there. Thus the \ 2-augmentations of ", Cell[BoxData[ \(TraditionalForm\`R\_72\)]], "should be in the complements of ", Cell[BoxData[ \(TraditionalForm\`S\_\(144 bis\)\)]], ". They are ", StyleBox["not", FontWeight->"Bold"], ", in the 2808 given by Vuza's algorithm. So I computed them and their \ orbits of 48 elements each." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(orbite[zoomIn[R72[\([1]\)], 2]]\)], "Input"], Cell[BoxData[ \({{1, 1, 1, 5, 1, 5, 1, 11, 1, 21, 1, 7, 1, 17, 1, 11, 1, 9, 1, 1, 1, 5, 1, 39}, {1, 1, 6, 1, 5, 12, 1, 12, 9, 1, 2, 5, 1, 17, 12, 10, 1, 1, 1, 5, 1, 6, 12, 21}, {1, 1, 6, 5, 1, 9, 2, 1, 5, 17, 8, 15, 2, 1, 5, 6, 1, 10, 1, 1, 6, 15, 8, 17}, {1, 2, 5, 1, 6, 11, 1, 11, 10, 1, 1, 6, 1, 18, 12, 9, 1, 1, 1, 5, 1, 5, 12, 22}, {1, 5, 1, 1, 1, 9, 1, 11, 1, 17, 1, 7, 1, 21, 1, 11, 1, 5, 1, 5, 1, 1, 1, 39}, {1, 5, 3, 3, 2, 5, 5, 7, 12, 6, 5, 3, 5, 10, 12, 7, 5, 1, 5, 1, 2, 5, 5, 29}, {1, 6, 1, 5, 2, 3, 7, 5, 12, 6, 7, 1, 7, 10, 12, 5, 6, 1, 5, 1, 1, 7, 3, 30}, {1, 7, 5, 10, 2, 1, 5, 7, 10, 2, 6, 15, 2, 6, 6, 11, 1, 1, 6, 6, 9, 3, 5, 17}, {1, 8, 3, 6, 1, 7, 5, 12, 6, 8, 3, 6, 6, 2, 10, 7, 6, 6, 2, 9, 1, 7, 10, 12}, {2, 6, 3, 2, 1, 5, 6, 1, 11, 11, 8, 3, 8, 7, 11, 1, 10, 1, 1, 6, 3, 2, 6, 29}, {2, 6, 5, 1, 1, 6, 5, 1, 12, 9, 8, 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15}, {2, 6, 6, 3, 2, 6, 1, 5, 12, 5, 8, 9, 8, 1, 12, 5, 5, 2, 5, 1, 9, 2, 6, 23}, {2, 6, 6, 5, 2, 5, 1, 6, 12, 3, 8, 11, 7, 1, 11, 7, 3, 2, 6, 1, 10, 2, 6, 21}, {3, 5, 5, 1, 1, 6, 5, 1, 11, 10, 8, 5, 8, 6, 11, 1, 9, 2, 1, 5, 5, 1, 7, 27}, {3, 8, 1, 10, 2, 5, 1, 11, 10, 2, 6, 11, 2, 6, 6, 12, 3, 2, 6, 6, 5, 7, 1, 18}, {5, 1, 2, 3, 2, 5, 5, 7, 5, 13, 5, 3, 5, 17, 5, 7, 5, 1, 5, 1, 2, 3, 2, 35}, {5, 1, 6, 1, 1, 9, 1, 11, 12, 6, 1, 7, 1, 10, 12, 11, 1, 5, 1, 5, 2, 1, 9, 25}, {5, 2, 1, 5, 6, 7, 5, 7, 10, 2, 3, 3, 5, 18, 12, 5, 2, 3, 2, 1, 5, 1, 12, 22}, {5, 3, 3, 6, 1, 1, 10, 1, 12, 6, 8, 3, 8, 10, 12, 1, 6, 5, 1, 2, 3, 7, 1, 29}, {5, 7, 1, 5, 5, 3, 9, 12, 6, 7, 1, 5, 6, 2, 10, 11, 6, 6, 2, 5, 5, 3, 10, 12}, {5, 7, 1, 9, 2, 6, 1, 12, 10, 2, 6, 9, 2, 6, 6, 12, 5, 2, 6, 6, 3, 8, 1, 17}, {6, 1, 1, 5, 2, 3, 7, 5, 7, 11, 7, 1, 7, 15, 7, 5, 6, 1, 5, 1, 1, 5, 2, 33}, {6, 1, 5, 2, 1, 9, 1, 12, 12, 5, 1, 7, 1, 9, 12, 12, 1, 5, 1, 6, 1, 1, 10, 23}, {6, 2, 9, 1, 5, 2, 5, 5, 12, 1, 8, 9, 8, 5, 12, 5, 1, 6, 2, 3, 6, 6, 2, 23}, {6, 2, 10, 1, 6, 2, 3, 7, 11, 1, 7, 11, 8, 3, 12, 6, 1, 5, 2, 5, 6, 6, 2, 21}, {6, 5, 1, 2, 3, 6, 1, 1, 10, 19, 8, 3, 7, 1, 10, 1, 11, 1, 6, 5, 3, 3, 8, 23}, {6, 6, 1, 1, 5, 5, 1, 2, 10, 17, 8, 5, 5, 3, 9, 1, 11, 1, 5, 7, 1, 5, 8, 21}, {6, 6, 2, 3, 7, 1, 10, 12, 7, 5, 3, 3, 7, 1, 11, 12, 6, 5, 3, 3, 6, 2, 10, 13}, {6, 6, 2, 9, 1, 5, 6, 2, 10, 7, 8, 10, 5, 7, 1, 10, 1, 6, 5, 3, 9, 6, 8, 11}, {6, 8, 9, 6, 6, 2, 10, 1, 6, 6, 2, 10, 5, 8, 10, 7, 5, 3, 9, 1, 5, 7, 1, 11}, {7, 1, 1, 6, 6, 5, 7, 5, 10, 2, 5, 1, 7, 18, 12, 3, 2, 5, 1, 1, 5, 1, 11, 22}, {7, 1, 5, 5, 1, 2, 9, 1, 11, 6, 8, 5, 8, 10, 11, 1, 5, 6, 1, 1, 5, 5, 3, 27}, {8, 3, 3, 5, 6, 1, 11, 1, 10, 1, 7, 3, 8, 19, 10, 1, 1, 6, 3, 2, 1, 5, 6, 23}, {8, 5, 1, 7, 5, 1, 11, 1, 9, 3, 5, 5, 8, 17, 10, 2, 1, 5, 5, 1, 1, 6, 6, 21}, {8, 5, 1, 9, 2, 1, 5, 6, 1, 10, 8, 15, 8, 6, 1, 10, 1, 1, 6, 5, 1, 9, 8, 17}, {8, 6, 9, 3, 5, 6, 1, 10, 1, 7, 5, 10, 8, 7, 10, 2, 6, 5, 1, 9, 2, 6, 6, 11}, {8, 9, 1, 5, 6, 1, 1, 10, 1, 6, 8, 15, 8, 10, 1, 6, 5, 1, 2, 9, 1, 5, 8, 17}, {8, 9, 2, 6, 1, 5, 7, 5, 5, 2, 6, 9, 8, 18, 5, 2, 5, 1, 6, 3, 2, 6, 1, 22}, {8, 11, 2, 5, 1, 6, 5, 7, 3, 2, 6, 11, 8, 18, 3, 2, 6, 1, 5, 5, 2, 5, 1, 21}, {9, 1, 2, 5, 1, 5, 1, 11, 12, 10, 1, 7, 1, 6, 12, 11, 1, 9, 1, 1, 6, 1, 5, 25}, {9, 3, 5, 5, 1, 7, 5, 12, 10, 3, 5, 5, 2, 6, 6, 11, 10, 2, 6, 5, 1, 7, 6, 12}, {10, 1, 1, 6, 1, 5, 1, 12, 12, 9, 1, 7, 1, 5, 12, 12, 1, 9, 1, 2, 5, 1, 6, 23}, {10, 2, 6, 3, 3, 5, 6, 12, 11, 1, 7, 3, 3, 5, 7, 12, 10, 1, 7, 3, 2, 6, 6, 13}, {11, 1, 5, 1, 1, 5, 2, 3, 12, 18, 7, 1, 5, 2, 10, 5, 7, 5, 6, 6, 1, 1, 7, 22}, {12, 1, 5, 1, 2, 3, 2, 5, 12, 18, 5, 3, 3, 2, 10, 7, 5, 7, 6, 5, 1, 2, 5, 22}, {12, 6, 5, 1, 2, 3, 2, 10, 12, 13, 5, 1, 2, 5, 5, 12, 5, 12, 1, 5, 3, 3, 2, 17}, {12, 6, 6, 1, 1, 5, 2, 10, 12, 11, 6, 1, 1, 7, 3, 12, 7, 11, 1, 6, 1, 5, 2, 15}}\)], "Output", GeneratedCell->False, CellAutoOverwrite->False] }, Closed]], Cell["Nothing new in the complements of these complements.", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["144 = 12\[Times]12", "Subsection"], Cell[TextData[{ "\[Bullet] Completing ", Cell[BoxData[ \(TraditionalForm\`R\_144\^1\)]], "(35 tiles of size 12 and period 144) gives 162 solutions, including the 6 \ given by Vuza's algorithm. Here they are (only the acyclic ones are given):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Select[%, \ aperQ]\)], "Input"], Cell[BoxData[ \({{2, 2, 26, 2, 4, 10, 20, 2, 14, 28, 2, 32}, {2, 2, 26, 6, 10, 20, 14, 2, 28, 2, 4, 28}, {2, 2, 28, 2, 2, 2, 28, 2, 2, 32, 10, 32}, {2, 2, 28, 4, 10, 22, 10, 4, 28, 2, 2, 30}, {2, 4, 22, 6, 4, 6, 20, 6, 10, 28, 6, 30}, {2, 4, 22, 10, 6, 20, 16, 2, 26, 6, 4, 26}, {2, 4, 26, 2, 2, 2, 26, 4, 2, 30, 14, 30}, {2, 4, 26, 4, 2, 26, 4, 14, 18, 14, 4, 26}, {2, 4, 28, 4, 6, 26, 6, 4, 28, 4, 2, 30}, {2, 14, 2, 14, 2, 2, 2, 14, 2, 14, 2, 74}, {2, 14, 14, 2, 4, 10, 6, 14, 2, 14, 28, 34}, {2, 14, 14, 6, 10, 6, 14, 14, 2, 28, 6, 28}, {2, 14, 16, 2, 2, 2, 14, 14, 2, 2, 32, 42}, {2, 14, 16, 4, 2, 14, 14, 2, 14, 20, 16, 26}, {2, 14, 16, 4, 10, 6, 16, 10, 4, 28, 4, 30}, {2, 14, 18, 2, 2, 14, 14, 4, 10, 22, 14, 28}, {2, 14, 20, 2, 14, 14, 14, 2, 20, 14, 2, 26}, {2, 14, 20, 10, 6, 26, 2, 2, 28, 4, 2, 28}, {2, 16, 10, 6, 4, 6, 10, 10, 6, 10, 28, 36}, {2, 16, 10, 10, 6, 10, 10, 16, 2, 26, 10, 26}, {2, 16, 14, 2, 2, 2, 16, 10, 4, 2, 30, 44}, {2, 16, 14, 4, 2, 16, 10, 4, 14, 18, 18, 26}, {2, 16, 14, 4, 10, 18, 4, 14, 18, 10, 16, 18}, {2, 16, 16, 2, 2, 16, 10, 6, 10, 20, 16, 28}, {2, 16, 16, 4, 6, 10, 16, 6, 4, 28, 4, 32}, {2, 16, 16, 4, 16, 16, 6, 16, 16, 4, 16, 16}, {2, 16, 18, 2, 16, 10, 16, 2, 18, 16, 2, 26}, {2, 16, 18, 10, 18, 16, 2, 18, 10, 6, 10, 18}, {2, 16, 20, 6, 10, 22, 4, 2, 26, 4, 6, 26}, {2, 26, 2, 14, 2, 18, 2, 14, 2, 26, 2, 34}, {2, 26, 6, 4, 6, 20, 2, 4, 10, 22, 6, 36}, {2, 26, 10, 6, 20, 2, 14, 2, 20, 6, 10, 26}, {2, 26, 16, 2, 18, 14, 2, 2, 26, 2, 4, 30}, {2, 28, 2, 4, 10, 18, 2, 2, 14, 18, 10, 34}, {2, 28, 6, 10, 18, 2, 14, 2, 18, 10, 6, 28}, {2, 28, 14, 2, 20, 10, 4, 2, 26, 2, 2, 32}, {2, 30, 2, 2, 2, 26, 2, 2, 2, 30, 2, 42}, {2, 30, 4, 10, 18, 4, 10, 4, 18, 10, 4, 30}, {2, 32, 4, 6, 22, 4, 6, 4, 22, 6, 4, 32}, {4, 2, 26, 2, 2, 14, 18, 2, 16, 26, 2, 30}, {4, 2, 26, 4, 2, 4, 26, 2, 4, 32, 6, 32}, {4, 6, 22, 4, 2, 16, 14, 6, 16, 22, 6, 26}, {4, 6, 22, 4, 18, 14, 18, 4, 22, 6, 4, 22}, {4, 10, 16, 2, 4, 10, 4, 16, 2, 14, 30, 32}, {4, 10, 18, 4, 6, 16, 10, 10, 16, 18, 10, 22}, {4, 10, 22, 6, 16, 20, 2, 14, 18, 4, 6, 22}, {4, 14, 14, 2, 2, 14, 4, 14, 2, 16, 26, 32}, {4, 14, 14, 4, 10, 16, 6, 14, 16, 14, 14, 18}, {4, 14, 14, 4, 14, 4, 14, 14, 4, 26, 6, 26}, {4, 14, 16, 2, 14, 4, 16, 10, 6, 26, 4, 28}, {4, 16, 10, 6, 10, 16, 4, 16, 16, 14, 16, 16}, {4, 16, 14, 2, 14, 16, 4, 16, 16, 10, 16, 16}, {4, 18, 10, 4, 2, 16, 4, 10, 6, 16, 22, 32}, {4, 18, 10, 4, 18, 4, 10, 18, 4, 22, 10, 22}, {4, 18, 14, 2, 16, 4, 16, 6, 10, 22, 4, 28}, {4, 22, 6, 4, 6, 16, 4, 6, 10, 16, 18, 32}, {4, 22, 10, 6, 16, 4, 16, 2, 14, 18, 4, 28}, {4, 26, 2, 4, 10, 16, 4, 2, 14, 16, 14, 32}, {4, 26, 6, 10, 16, 4, 14, 2, 16, 14, 4, 28}, {4, 28, 2, 2, 14, 16, 2, 2, 16, 16, 10, 32}, {4, 28, 4, 2, 16, 14, 2, 4, 16, 16, 6, 32}, {4, 28, 4, 6, 16, 10, 6, 4, 16, 16, 2, 32}, {4, 28, 4, 10, 16, 6, 10, 4, 16, 14, 2, 30}, {6, 4, 18, 14, 2, 20, 16, 6, 22, 10, 4, 22}, {6, 4, 22, 4, 2, 4, 22, 4, 6, 26, 18, 26}, {6, 10, 6, 10, 4, 2, 4, 10, 6, 10, 6, 70}, {6, 10, 16, 2, 2, 14, 2, 16, 2, 16, 28, 30}, {6, 10, 16, 4, 2, 4, 10, 16, 2, 4, 32, 38}, {6, 10, 16, 4, 6, 10, 16, 2, 16, 20, 16, 22}, {6, 10, 20, 2, 4, 10, 18, 4, 6, 26, 10, 28}, {6, 10, 20, 6, 10, 18, 10, 6, 20, 10, 6, 22}, {6, 14, 14, 2, 14, 14, 6, 16, 14, 14, 14, 16}, {6, 16, 6, 10, 4, 2, 14, 6, 10, 6, 28, 36}, {6, 16, 6, 14, 2, 14, 6, 16, 6, 22, 14, 22}, {6, 16, 10, 4, 2, 4, 16, 6, 4, 6, 26, 44}, {6, 16, 10, 4, 6, 16, 6, 4, 18, 14, 22, 22}, {6, 16, 14, 2, 4, 16, 6, 10, 6, 20, 16, 28}, {6, 16, 14, 6, 16, 6, 16, 6, 14, 16, 6, 22}, {6, 16, 16, 4, 2, 14, 16, 2, 4, 28, 4, 32}, {6, 22, 6, 10, 6, 14, 6, 10, 6, 22, 6, 30}, {6, 22, 10, 4, 2, 20, 6, 4, 6, 26, 2, 36}, {6, 22, 14, 2, 20, 6, 10, 6, 20, 2, 14, 22}, {6, 22, 16, 6, 14, 16, 2, 4, 22, 6, 4, 26}, {6, 26, 2, 2, 14, 14, 4, 2, 16, 14, 14, 30}, {6, 26, 4, 2, 4, 22, 4, 2, 4, 26, 6, 38}, {6, 28, 10, 6, 20, 6, 4, 6, 22, 4, 2, 30}, {10, 4, 14, 16, 2, 18, 16, 10, 18, 14, 4, 18}, {10, 4, 18, 4, 6, 4, 18, 4, 10, 22, 22, 22}, {10, 6, 10, 6, 4, 6, 4, 6, 10, 6, 10, 66}, {10, 6, 16, 4, 6, 4, 6, 16, 6, 4, 32, 34}, {10, 16, 2, 14, 2, 2, 16, 2, 14, 2, 28, 36}, {10, 16, 2, 16, 2, 16, 2, 16, 10, 18, 18, 18}, {10, 16, 6, 4, 6, 4, 16, 2, 4, 10, 22, 44}, {10, 16, 10, 6, 4, 16, 2, 14, 2, 20, 16, 28}, {10, 16, 10, 10, 16, 2, 16, 10, 10, 16, 10, 18}, {10, 16, 16, 2, 2, 16, 14, 2, 2, 28, 4, 32}, {10, 18, 10, 6, 10, 10, 10, 6, 10, 18, 10, 26}, {10, 18, 14, 2, 2, 18, 10, 4, 2, 28, 2, 34}, {10, 18, 16, 10, 10, 16, 6, 4, 18, 10, 4, 22}, {10, 22, 4, 2, 16, 10, 4, 6, 16, 10, 18, 26}, {10, 22, 4, 6, 4, 18, 4, 6, 4, 22, 10, 34}, {10, 26, 6, 4, 18, 10, 4, 2, 20, 10, 6, 28}, {14, 2, 14, 2, 4, 10, 4, 2, 14, 2, 14, 62}, {14, 14, 2, 14, 2, 4, 14, 2, 14, 2, 26, 36}, {14, 14, 14, 2, 14, 6, 14, 2, 14, 14, 14, 22}, {14, 14, 16, 2, 4, 14, 14, 2, 2, 26, 6, 30}, {14, 14, 16, 14, 6, 16, 10, 4, 14, 14, 4, 18}, {14, 16, 2, 4, 10, 4, 14, 2, 2, 14, 18, 44}, {14, 16, 6, 10, 4, 14, 2, 14, 2, 18, 16, 28}, {14, 16, 14, 2, 4, 16, 10, 4, 2, 26, 4, 32}, {14, 18, 4, 6, 16, 6, 4, 10, 16, 6, 22, 22}, {14, 18, 4, 10, 4, 14, 4, 10, 4, 18, 14, 30}, {14, 22, 10, 4, 14, 14, 2, 2, 18, 14, 2, 28}, {16, 2, 14, 2, 2, 14, 2, 2, 14, 2, 16, 58}, {16, 6, 10, 4, 2, 14, 2, 4, 10, 6, 16, 54}, {16, 6, 14, 2, 14, 2, 4, 16, 6, 10, 26, 28}, {16, 6, 16, 6, 10, 4, 16, 6, 10, 18, 16, 20}, {16, 10, 6, 4, 6, 10, 6, 4, 6, 10, 16, 50}, {16, 10, 10, 6, 10, 6, 4, 16, 2, 14, 22, 28}, {16, 10, 16, 2, 14, 4, 16, 2, 14, 14, 16, 20}, {16, 14, 2, 4, 10, 6, 10, 4, 2, 14, 16, 46}, {16, 14, 6, 10, 6, 10, 4, 14, 2, 16, 18, 28}, {16, 14, 14, 2, 16, 4, 14, 2, 16, 10, 16, 20}, {16, 16, 2, 2, 14, 2, 14, 2, 2, 16, 16, 42}, {16, 16, 4, 2, 14, 2, 14, 2, 4, 16, 16, 38}, {16, 16, 4, 6, 10, 6, 10, 6, 4, 16, 16, 34}, {16, 16, 4, 10, 6, 10, 6, 10, 4, 16, 16, 30}, {16, 16, 4, 14, 2, 14, 2, 14, 4, 16, 16, 26}, {16, 16, 4, 16, 2, 14, 2, 16, 4, 16, 16, 22}, {16, 16, 4, 16, 6, 10, 6, 16, 4, 16, 16, 18}, {16, 18, 2, 14, 2, 14, 4, 10, 6, 16, 14, 28}, {16, 18, 10, 6, 16, 4, 10, 6, 16, 6, 16, 20}, {16, 20, 2, 14, 2, 16, 4, 6, 10, 16, 10, 28}, {16, 20, 6, 10, 6, 16, 4, 2, 14, 16, 6, 28}, {16, 20, 10, 6, 10, 16, 2, 2, 16, 16, 2, 28}, {16, 20, 14, 2, 14, 14, 2, 4, 16, 14, 2, 26}, {16, 20, 16, 2, 16, 10, 6, 4, 16, 10, 6, 22}, {18, 10, 6, 10, 6, 4, 10, 6, 10, 6, 22, 36}, {18, 10, 16, 6, 4, 10, 16, 2, 4, 22, 10, 26}, {18, 14, 2, 2, 14, 4, 10, 4, 2, 16, 14, 44}, {18, 14, 4, 14, 4, 10, 4, 14, 4, 14, 18, 26}, {18, 16, 2, 14, 4, 10, 6, 10, 6, 14, 16, 28}, {18, 16, 10, 6, 4, 16, 6, 4, 6, 22, 4, 32}, {18, 18, 14, 4, 10, 16, 2, 4, 14, 16, 2, 26}, {22, 6, 10, 6, 10, 4, 6, 10, 6, 10, 18, 36}, {22, 10, 4, 2, 16, 4, 6, 4, 6, 16, 10, 44}, {22, 14, 2, 16, 4, 6, 10, 6, 10, 10, 16, 28}, {22, 16, 6, 10, 4, 16, 2, 4, 10, 18, 4, 32}, {26, 2, 14, 2, 14, 4, 2, 14, 2, 14, 14, 36}, {26, 6, 4, 6, 16, 4, 2, 4, 10, 16, 6, 44}, {26, 10, 6, 16, 4, 2, 14, 2, 14, 6, 16, 28}, {26, 16, 2, 14, 4, 14, 2, 2, 14, 14, 4, 32}, {28, 2, 14, 2, 16, 2, 2, 14, 2, 16, 10, 36}, {28, 6, 10, 6, 14, 2, 4, 10, 6, 16, 6, 36}, {28, 10, 6, 10, 10, 6, 4, 6, 10, 16, 2, 36}, {28, 14, 2, 14, 6, 10, 4, 2, 14, 14, 2, 34}, {28, 16, 2, 16, 2, 14, 2, 2, 16, 10, 6, 30}, {30, 2, 4, 10, 16, 2, 2, 2, 14, 16, 2, 44}, {30, 14, 2, 16, 4, 10, 4, 2, 16, 10, 4, 32}, {32, 2, 2, 14, 14, 2, 2, 2, 16, 14, 2, 42}, {32, 4, 2, 16, 10, 4, 2, 4, 16, 10, 6, 38}, {32, 4, 6, 16, 6, 4, 6, 4, 16, 6, 10, 34}}\)], "Output", GeneratedCell->False, CellAutoOverwrite->False] }, Closed]], Cell["32 of them are palindroms.", "Text"], Cell[TextData[{ "\[Bullet] Doubling ", Cell[BoxData[ \(TraditionalForm\`R\_72\)]], "(the dual augmentation from the preceding case) one gets 6 tiles of length \ 12, and period 144. They have 324 complements, all of which are acyclic, all \ new, for 6\[Times]324=1944 new canons. Here comes:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(complete[{2, 6, 6, 12, 22, 8, 18, 12, 10, 2, 6, 40}]\)], "Input"], Cell[BoxData[ \({{1, 3, 25, 3, 4, 9, 20, 3, 13, 28, 3, 32}, {1, 3, 25, 7, 9, 20, 15, 1, 28, 3, 4, 28}, {1, 3, 28, 1, 3, 1, 28, 3, 1, 32, 11, 32}, {1, 3, 28, 4, 9, 23, 9, 4, 28, 3, 1, 31}, {1, 4, 23, 5, 4, 7, 20, 5, 11, 28, 5, 31}, {1, 4, 23, 9, 7, 20, 16, 1, 27, 5, 4, 27}, {1, 4, 27, 1, 3, 1, 27, 4, 1, 31, 13, 31}, {1, 4, 27, 4, 1, 27, 4, 13, 19, 13, 4, 27}, {1, 4, 28, 3, 1, 27, 5, 11, 20, 13, 3, 28}, {1, 4, 28, 4, 7, 25, 7, 4, 28, 4, 1, 31}, {1, 15, 1, 15, 1, 3, 1, 15, 1, 15, 1, 75}, {1, 15, 13, 3, 4, 9, 7, 13, 3, 13, 28, 35}, {1, 15, 13, 7, 9, 7, 13, 15, 1, 28, 7, 28}, {1, 15, 16, 1, 3, 1, 15, 13, 3, 1, 32, 43}, {1, 15, 16, 4, 1, 15, 13, 3, 13, 20, 16, 27}, {1, 15, 16, 4, 9, 7, 16, 9, 4, 28, 4, 31}, {1, 15, 17, 3, 1, 15, 13, 4, 11, 21, 15, 28}, {1, 15, 20, 1, 15, 13, 15, 1, 20, 15, 1, 27}, {1, 15, 20, 9, 7, 25, 3, 1, 28, 4, 3, 28}, {1, 16, 11, 5, 4, 7, 9, 11, 5, 11, 28, 36}, {1, 16, 11, 9, 7, 9, 11, 16, 1, 27, 9, 27}, {1, 16, 15, 1, 3, 1, 16, 11, 4, 1, 31, 44}, {1, 16, 15, 4, 1, 16, 11, 4, 13, 19, 17, 27}, {1, 16, 15, 4, 9, 19, 4, 13, 19, 9, 16, 19}, {1, 16, 16, 3, 1, 16, 11, 5, 11, 20, 16, 28}, {1, 16, 16, 4, 7, 9, 16, 7, 4, 28, 4, 32}, {1, 16, 16, 4, 16, 16, 7, 16, 16, 4, 16, 16}, {1, 16, 19, 1, 16, 11, 16, 1, 19, 16, 1, 27}, {1, 16, 19, 9, 19, 16, 1, 19, 9, 7, 9, 19}, {1, 16, 20, 7, 9, 23, 4, 1, 27, 4, 5, 27}, {1, 27, 1, 15, 1, 19, 1, 15, 1, 27, 1, 35}, {1, 27, 5, 4, 7, 20, 1, 4, 11, 21, 7, 36}, {1, 27, 9, 7, 20, 1, 15, 1, 20, 7, 9, 27}, {1, 27, 16, 1, 19, 13, 3, 1, 27, 1, 4, 31}, {1, 28, 3, 4, 9, 19, 1, 3, 13, 19, 9, 35}, {1, 28, 7, 9, 19, 1, 15, 1, 19, 9, 7, 28}, {1, 28, 15, 1, 20, 11, 4, 1, 27, 1, 3, 32}, {1, 31, 1, 3, 1, 27, 1, 3, 1, 31, 1, 43}, {1, 31, 4, 9, 19, 4, 9, 4, 19, 9, 4, 31}, {1, 32, 4, 7, 21, 4, 7, 4, 21, 7, 4, 32}, {3, 1, 27, 1, 4, 11, 20, 1, 15, 28, 1, 32}, {3, 1, 28, 3, 1, 3, 28, 1, 3, 32, 9, 32}, {3, 1, 28, 4, 11, 21, 11, 4, 28, 1, 3, 29}, {3, 4, 21, 7, 4, 5, 20, 7, 9, 28, 7, 29}, {3, 4, 21, 11, 5, 20, 16, 3, 25, 7, 4, 25}, {3, 4, 25, 3, 1, 3, 25, 4, 3, 29, 15, 29}, {3, 4, 25, 4, 3, 25, 4, 15, 17, 15, 4, 25}, {3, 4, 28, 4, 5, 27, 5, 4, 28, 4, 3, 29}, {3, 13, 3, 13, 3, 1, 3, 13, 3, 13, 3, 73}, {3, 13, 15, 1, 4, 11, 5, 15, 1, 15, 28, 33}, {3, 13, 15, 5, 11, 5, 15, 13, 3, 28, 5, 28}, {3, 13, 16, 3, 1, 3, 13, 15, 1, 3, 32, 41}, {3, 13, 16, 4, 3, 13, 15, 1, 15, 20, 16, 25}, {3, 13, 16, 4, 11, 5, 16, 11, 4, 28, 4, 29}, {3, 13, 19, 1, 3, 13, 15, 4, 9, 23, 13, 28}, {3, 13, 20, 3, 13, 15, 13, 3, 20, 13, 3, 25}, {3, 13, 20, 11, 5, 27, 1, 3, 28, 4, 1, 28}, {3, 16, 9, 7, 4, 5, 11, 9, 7, 9, 28, 36}, {3, 16, 9, 11, 5, 11, 9, 16, 3, 25, 11, 25}, {3, 16, 13, 3, 1, 3, 16, 9, 4, 3, 29, 44}, {3, 16, 13, 4, 3, 16, 9, 4, 15, 17, 19, 25}, {3, 16, 13, 4, 11, 17, 4, 15, 17, 11, 16, 17}, {3, 16, 16, 1, 3, 16, 9, 7, 9, 20, 16, 28}, {3, 16, 16, 4, 5, 11, 16, 5, 4, 28, 4, 32}, {3, 16, 16, 4, 16, 16, 5, 16, 16, 4, 16, 16}, 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19, 4, 5, 27, 9, 28}, {7, 9, 20, 7, 9, 19, 9, 7, 20, 9, 7, 21}, {7, 13, 15, 1, 15, 13, 7, 16, 13, 15, 13, 16}, {7, 16, 5, 11, 4, 1, 15, 5, 11, 5, 28, 36}, {7, 16, 5, 15, 1, 15, 5, 16, 7, 21, 15, 21}, {7, 16, 9, 4, 3, 4, 16, 5, 4, 7, 25, 44}, {7, 16, 13, 3, 4, 16, 5, 11, 5, 20, 16, 28}, {7, 16, 13, 7, 16, 5, 16, 7, 13, 16, 7, 21}, {7, 16, 16, 4, 1, 15, 16, 1, 4, 28, 4, 32}, {7, 21, 7, 9, 7, 13, 7, 9, 7, 21, 7, 29}, {7, 21, 11, 4, 1, 20, 7, 4, 5, 27, 1, 36}, {7, 21, 15, 1, 20, 7, 9, 7, 20, 1, 15, 21}, {7, 21, 16, 7, 13, 16, 3, 4, 21, 7, 4, 25}, {7, 25, 3, 1, 15, 13, 4, 3, 16, 13, 15, 29}, {7, 25, 4, 3, 4, 21, 4, 3, 4, 25, 7, 37}, {7, 28, 9, 7, 20, 5, 4, 7, 21, 4, 3, 29}, {9, 4, 15, 16, 1, 19, 16, 9, 19, 13, 4, 19}, {9, 4, 19, 4, 5, 4, 19, 4, 9, 23, 21, 23}, {9, 7, 9, 7, 4, 5, 4, 7, 9, 7, 9, 67}, {9, 7, 16, 4, 5, 4, 7, 16, 5, 4, 32, 35}, {9, 16, 3, 13, 3, 1, 16, 3, 13, 3, 28, 36}, {9, 16, 3, 16, 1, 16, 3, 16, 9, 19, 17, 19}, {9, 16, 7, 4, 5, 4, 16, 3, 4, 9, 23, 44}, {9, 16, 11, 5, 4, 16, 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13, 3, 1, 15, 1, 3, 13, 3, 16, 57}, {16, 5, 11, 4, 1, 15, 1, 4, 11, 5, 16, 55}, {16, 5, 15, 1, 15, 1, 4, 16, 7, 9, 27, 28}, {16, 5, 16, 7, 9, 4, 16, 7, 9, 19, 16, 20}, {16, 7, 9, 4, 3, 13, 3, 4, 9, 7, 16, 53}, {16, 7, 13, 3, 13, 3, 4, 16, 5, 11, 25, 28}, {16, 7, 16, 5, 11, 4, 16, 5, 11, 17, 16, 20}, {16, 9, 7, 4, 5, 11, 5, 4, 7, 9, 16, 51}, {16, 9, 11, 5, 11, 5, 4, 16, 3, 13, 23, 28}, {16, 9, 16, 3, 13, 4, 16, 3, 13, 15, 16, 20}, {16, 11, 5, 4, 7, 9, 7, 4, 5, 11, 16, 49}, {16, 11, 9, 7, 9, 7, 4, 16, 1, 15, 21, 28}, {16, 11, 16, 1, 15, 4, 16, 1, 15, 13, 16, 20}, {16, 13, 3, 4, 9, 7, 9, 4, 3, 13, 16, 47}, {16, 13, 7, 9, 7, 9, 4, 15, 1, 16, 19, 28}, {16, 13, 15, 1, 16, 4, 15, 1, 16, 11, 16, 20}, {16, 15, 1, 4, 11, 5, 11, 4, 1, 15, 16, 45}, {16, 15, 5, 11, 5, 11, 4, 13, 3, 16, 17, 28}, {16, 15, 13, 3, 16, 4, 13, 3, 16, 9, 16, 20}, {16, 16, 1, 3, 13, 3, 13, 3, 1, 16, 16, 43}, {16, 16, 3, 1, 15, 1, 15, 1, 3, 16, 16, 41}, {16, 16, 4, 1, 15, 1, 15, 1, 4, 16, 16, 39}, {16, 16, 4, 3, 13, 3, 13, 3, 4, 16, 16, 37}, {16, 16, 4, 5, 11, 5, 11, 5, 4, 16, 16, 35}, {16, 16, 4, 7, 9, 7, 9, 7, 4, 16, 16, 33}, {16, 16, 4, 9, 7, 9, 7, 9, 4, 16, 16, 31}, {16, 16, 4, 11, 5, 11, 5, 11, 4, 16, 16, 29}, {16, 16, 4, 13, 3, 13, 3, 13, 4, 16, 16, 27}, {16, 16, 4, 15, 1, 15, 1, 15, 4, 16, 16, 25}, {16, 16, 4, 16, 1, 15, 1, 16, 4, 16, 16, 23}, {16, 16, 4, 16, 3, 13, 3, 16, 4, 16, 16, 21}, {16, 16, 4, 16, 5, 11, 5, 16, 4, 16, 16, 19}, {16, 16, 4, 16, 7, 9, 7, 16, 4, 16, 16, 17}, {16, 17, 3, 13, 3, 13, 4, 11, 5, 16, 15, 28}, {16, 17, 11, 5, 16, 4, 11, 5, 16, 7, 16, 20}, {16, 19, 1, 15, 1, 15, 4, 9, 7, 16, 13, 28}, {16, 19, 9, 7, 16, 4, 9, 7, 16, 5, 16, 20}, {16, 20, 1, 15, 1, 16, 4, 7, 9, 16, 11, 28}, {16, 20, 3, 13, 3, 16, 4, 5, 11, 16, 9, 28}, {16, 20, 5, 11, 5, 16, 4, 3, 13, 16, 7, 28}, {16, 20, 7, 9, 7, 16, 4, 1, 15, 16, 5, 28}, {16, 20, 9, 7, 9, 16, 3, 1, 16, 16, 3, 28}, {16, 20, 11, 5, 11, 16, 1, 3, 16, 16, 1, 28}, {16, 20, 13, 3, 13, 15, 1, 4, 16, 15, 1, 27}, {16, 20, 15, 1, 15, 13, 3, 4, 16, 13, 3, 25}, {16, 20, 16, 1, 16, 11, 5, 4, 16, 11, 5, 23}, {16, 20, 16, 3, 16, 9, 7, 4, 16, 9, 7, 21}, {17, 11, 5, 11, 5, 4, 11, 5, 11, 5, 23, 36}, {17, 11, 16, 1, 16, 3, 16, 1, 16, 11, 17, 19}, {17, 11, 16, 5, 4, 11, 16, 1, 4, 23, 9, 27}, {17, 15, 1, 3, 13, 4, 11, 4, 1, 16, 15, 44}, {17, 15, 4, 13, 4, 11, 4, 13, 4, 15, 17, 27}, {17, 16, 3, 13, 4, 11, 5, 11, 5, 15, 16, 28}, {17, 16, 11, 5, 4, 16, 7, 4, 5, 23, 4, 32}, {17, 19, 13, 4, 11, 16, 1, 4, 15, 16, 1, 27}, {19, 9, 7, 9, 7, 4, 9, 7, 9, 7, 21, 36}, {19, 9, 16, 7, 4, 9, 16, 3, 4, 21, 11, 25}, {19, 13, 3, 1, 15, 4, 9, 4, 3, 16, 13, 44}, {19, 13, 4, 15, 4, 9, 4, 15, 4, 13, 19, 25}, {19, 16, 1, 15, 4, 9, 7, 9, 7, 13, 16, 28}, {19, 16, 9, 7, 4, 16, 5, 4, 7, 21, 4, 32}, {19, 17, 15, 4, 9, 16, 3, 4, 13, 16, 3, 25}, {21, 7, 9, 7, 9, 4, 7, 9, 7, 9, 19, 36}, {21, 7, 16, 9, 4, 7, 16, 5, 4, 19, 13, 23}, {21, 11, 4, 1, 16, 4, 7, 4, 5, 16, 11, 44}, {21, 11, 4, 17, 4, 7, 4, 17, 4, 11, 21, 23}, {21, 15, 1, 16, 4, 7, 9, 7, 9, 11, 16, 28}, {21, 15, 17, 4, 7, 16, 5, 4, 11, 16, 5, 23}, {21, 16, 7, 9, 4, 16, 3, 4, 9, 19, 4, 32}, {23, 5, 11, 5, 11, 4, 5, 11, 5, 11, 17, 36}, {23, 9, 4, 3, 16, 4, 5, 4, 7, 16, 9, 44}, {23, 13, 3, 16, 4, 5, 11, 5, 11, 9, 16, 28}, {23, 16, 5, 11, 4, 16, 1, 4, 11, 17, 4, 32}, {25, 3, 13, 3, 13, 4, 3, 13, 3, 13, 15, 36}, {25, 7, 4, 5, 16, 4, 3, 4, 9, 16, 7, 44}, {25, 11, 5, 16, 4, 3, 13, 3, 13, 7, 16, 28}, {25, 16, 3, 13, 4, 15, 1, 3, 13, 15, 4, 32}, {27, 1, 15, 1, 15, 4, 1, 15, 1, 15, 13, 36}, {27, 5, 4, 7, 16, 4, 1, 4, 11, 16, 5, 44}, {27, 9, 7, 16, 4, 1, 15, 1, 15, 5, 16, 28}, {27, 16, 1, 15, 4, 13, 3, 1, 15, 13, 4, 32}, {28, 1, 15, 1, 16, 3, 1, 15, 1, 16, 11, 36}, {28, 3, 13, 3, 16, 1, 3, 13, 3, 16, 9, 36}, {28, 5, 11, 5, 15, 1, 4, 11, 5, 16, 7, 36}, {28, 7, 9, 7, 13, 3, 4, 9, 7, 16, 5, 36}, {28, 9, 7, 9, 11, 5, 4, 7, 9, 16, 3, 36}, {28, 11, 5, 11, 9, 7, 4, 5, 11, 16, 1, 36}, {28, 13, 3, 13, 7, 9, 4, 3, 13, 15, 1, 35}, {28, 15, 1, 15, 5, 11, 4, 1, 15, 13, 3, 33}, {28, 16, 1, 16, 3, 13, 3, 1, 16, 11, 5, 31}, {28, 16, 3, 16, 1, 15, 1, 3, 16, 9, 7, 29}, {29, 3, 4, 9, 16, 3, 1, 3, 13, 16, 3, 44}, {29, 15, 1, 16, 4, 11, 4, 1, 16, 11, 4, 32}, {31, 1, 4, 11, 16, 1, 3, 1, 15, 16, 1, 44}, {31, 13, 3, 16, 4, 9, 4, 3, 16, 9, 4, 32}, {32, 1, 3, 13, 15, 1, 3, 1, 16, 15, 1, 43}, {32, 3, 1, 15, 13, 3, 1, 3, 16, 13, 3, 41}, {32, 4, 1, 16, 11, 4, 1, 4, 16, 11, 5, 39}, {32, 4, 3, 16, 9, 4, 3, 4, 16, 9, 7, 37}, {32, 4, 5, 16, 7, 4, 5, 4, 16, 7, 9, 35}, {32, 4, 7, 16, 5, 4, 7, 4, 16, 5, 11, 33}}\)], "Output", GeneratedCell->False, CellAutoOverwrite->False] }, Closed]], Cell[TextData[{ "Complementation ot these gives 2 ", Cell[BoxData[ \(TraditionalForm\`S\_72 = \ S\_144\^bis\)]], ", as expected, and nothing more." }], "Text"], Cell[TextData[{ "\[Bullet] I cannot use ", StyleBox["complete", "Input"], " on my computer for the 3 elements of but with ", StyleBox["zoomOut", "Input"], " and ", StyleBox["orbite", "Input"], " I found a few\[LongDash] 432 \[LongDash] of the colmplements, that is to \ say ", Cell[BoxData[ \(TraditionalForm\`3\[Times]432 = 1296\)]], " more." }], "Text"], Cell[CellGroupData[{ Cell["Pour 108 :", "Subsubsection"], Cell[TextData[{ "En ", StyleBox["compl\[EAcute]tant", FontWeight->"Bold", FontSlant->"Italic"], ", tout simplement, le ", Cell[BoxData[ \(TraditionalForm\`S\_108\)]], "de l'algorithme, on trouve 72 nouvelles fa\[CCedilla]ons en plus des 1780 \ donn\[EAcute]es par l'algorithme (soit 216 nouveaux canons). Il resterait \ \[AGrave] tenter de compl\[EAcute]ter des motifs acycliques cousins des Vuza \ canons connus, obtenus comme compl\[EAcute]ments des compl\[EAcute]ments de \ ", Cell[BoxData[ \(TraditionalForm\`S\_108\)]], ", comme {3,3,21,3,3,75} (mais c'est long !). Ceci dit cela fait \ d\[EAcute]j\[AGrave] 45 motifs non dans ", Cell[BoxData[ \(TraditionalForm\`S\_108\)]], ", qui compl\[EGrave]tent les 252 trouv\[EAcute]s soit 11340 nouveaux \ canons de plus !" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Pour 120 :", "Subsubsection"], Cell[TextData[{ "En quintuplant ", Cell[BoxData[ \(TraditionalForm\`R\_72\)]], " et divisant par 2, on obtient trois motifs de p\[EAcute]riode 120 et de \ longueur 6. Une recherche al\[EAcute]atoire donne moult compl\[EAcute]ments, \ mais je n'ai pas la puissance n\[EAcute]cessaire pour les trouver tous; \ remarquablement, tous ces compl\[EAcute]ments sont acycliques ! Pour \ l'instant cela donne 3\[Times]239 = 717 nouveaux canons, mais on sait que \ c'est incomplet." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["120", "Subsection"], Cell[TextData[{ "Completion of the Vuza canons (", Cell[BoxData[ \(TraditionalForm\`r\_120\)]], ") gives nothing new. One strange thing : a lone complement has inner \ periodicity, all others are acyclic. The funny one is {10,30,10,30,10,30}, \ the others form two orbits of four. I am not able to complete ", Cell[BoxData[ \(TraditionalForm\`S\_120\)]], "for lack of computer power.\nSo I found another track : compute ", Cell[BoxData[ \(S72*5/2\)], "Input"], " which gives length 6, period 120 tiles. The random method gives a few of \ the complements ; members of their orbits are " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(%\)], "Input"], Cell["\<\ {{1, 1, 1, 7, 1, 6, 1, 14, 1, 6, 15, 6, 1, 8, 1, 1, 21, 1, 14, 1, \ 12, 1, 1, 1, 6, 1, 1, 13, 15, 21}, {1, 1, 2, 7, 3, 1, 2, 13, 2, 6, 15, 8, 3, 4, 3, 3, 16, 2, 13, 2, 13, 1, 1, 2, 4, 3, 3, 9, 15, 22}, {1, 1, 2, 7, 3, 3, 15, 6, 15, 7, 1, 3, 4, 3, 3, 1, 15, 2, 15, 14, 1, 2, 4, 3, 3, 9, 7, 8, 7, 15}, {1, 1, 6, 2, 1, 7, 1, 13, 2, 15, 11, 1, 1, 6, 2, 1, \ 7, 14, 17, 11, 1, 1, 2, 4, 2, 1, 7, 1, 13, 28}, {1, 1, 7, 1, 1, 6, 7, 8, 6, 1, 14, 7, 1, 7, 2, 1, 21, 7, 8, 7, 6, 1, 1, 6, 1, 1, 1, 12, 15, 22}, {1, 1, 7, 3, 3, 1, 3, 9, 2, 1, 12, 15, 4, 7, 3, 7, 11, 1, 14, 1, 12, 2, 1, 1, 7, 3, 1, 6, 9, 32}, {1, 1, 7, 3, 4, 3, 9, 3, 12, 15, 2, 2, 7, 3, 3, 4, 11, 1, 15, 12, 3, 1, 7, 3, 1, 6, 9, 2, 15, 15}, {1, 1, 8, 1, 6, 15, 6, 1, 14, 1, 6, 1, 7, 1, 1, 1, \ 21, 15, 13, 1, 1, 6, 1, 1, 1, 12, 1, 14, 1, 21}, {1, 1, 8, 6, 1, 12, 2, 1, 6, 1, 14, 7, 8, 2, 19, 2, 1, 7, 7, 1, 13, 1, 1, 6, 2, 13, 6, 9, 1, 21}, {1, 1, 8, 6, 3, 4, 8, 1, 1, 1, 15, 11, 2, 8, 13, 8, 1, 1, 13, 3, 11, 1, 1, 2, 6, 9, 4, 9, 1, 27}, {1, 1, 8, 7, 12, 3, 6, 1, 14, 7, 1, 7, 2, 6, 13, 2, 1, 7, 8, 13, 2, 6, 2, 13, 6, 2, 7, 1, 7, 14}, {1, 1, 9, 4, 2, 6, 11, 4, 6, 9, 8, 10, 4, 2, 13, 2, 2, 10, 1, 2, 6, 8, 7, 3, 12, 11, 10, 1, 2, 13}, {1, 1, 9, 6, 15, 2, 4, 6, 9, 7, 1, 7, 3, 4, 15, 2, 2, 10, 3, 14, 1, 6, 3, 12, 7, 4, 4, 6, 1, 15}, {1, 2, 4, 6, 2, 1, 14, 1, 3, 3, 12, 14, 1, 3, 6, 17, 1, 6, 8, 1, 14, 1, 2, 1, 3, 6, 6, 15, 3, 23}, {1, 2, 4, 6, 3, 15, 3, 3, 12, 11, 3, 1, 3, 6, 2, 15, 1, 6, 9, 15, 2, 1, 3, 6, 6, 11, 4, 3, 8, 15}, {1, 2, 7, 3, 3, 13, 2, 6, 7, 8, 7, 1, 3, 4, 3, 3, 18, 15, 13, 1, 1, 2, 4, 3, 3, 1, 8, 7, 8, 23}, {1, 2, 9, 3, 1, 11, 4, 8, 3, 12, 9, 6, 6, 3, 12, 1, 2, 9, 3, 1, 11, 4, 8, 3, 12, 9, 6, 6, 3, 12}, {1, 2, 9, 3, 12, 12, 3, 12, 7, 2, 6, 6, 1, 2, 12, 1, 2, 9, 3, 12, 12, 3, 12, 7, 2, 6, 6, 1, 2, 12}, {1, 3, 3, 4, 3, 1, 7, 15, 6, 15, 3, 3, 7, 2, 1, 1, 15, 7, 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1, 3, 10, 1, 3, 12, 3, 14, 1, 3, \ 10, 1, 2, 10, 3, 13, 15}, {3, 3, 9, 1, 2, 9, 2, 4, 4, 7, 15, 7, 9, 1, 11, 3, 3, 4, 8, 3, 11, 4, 3, 8, 2, 11, 2, 8, 8, 15}, {3, 4, 3, 4, 3, 1, 3, 8, 4, 3, 8, 7, 9, 7, 3, 7, 16, 11, 4, 3, 8, 1, 3, 3, 1, 3, 4, 3, 4, 39}, {3, 4, 3, 6, 2, 4, 7, 2, 2, 4, 2, 7, 14, 10, 19, 4, 4, \ 2, 7, 2, 4, 8, 1, 2, 4, 3, 6, 13, 10, 21}, {3, 4, 3, 7, 1, 3, 12, 3, 8, 7, \ 8, 1, 7, 3, 4, 3, 16, 15, 3, 9, 3, 3, 1, 3, 4, 3, 4, 8, 15, 16}, {3, 4, 3, 7, 1, 8, 7, 8, 3, 12, 3, 1, 7, 3, 4, 3, 16, 15, 8, 4, 3, 4, 3, 1, \ 3, 3, 9, 3, 15, 16}, {3, 4, 6, 1, 1, 1, 6, 2, 6, 1, 6, 9, 17, 10, 1, 10, 6, 7, 8, 1, 6, 8, 1, 2, 4, 6, 1, 2, 8, 36}, {3, 4, 6, 1, 2, 6, 2, 7, 6, 9, \ 14, 3, 10, 1, 1, 9, 6, 7, 9, 6, 9, 2, 4, 6, 1, 2, 8, 6, 15, 15}, {3, 4, 11, 4, 2, 6, 3, 1, 2, 1, 14, 9, 6, 18, 6, 3, 1, 3, 11, 4, 8, 3, 3, 1, 14, 3, 6, 4, 2, 24}, {3, 6, 1, 7, 1, 1, 2, 11, 1, 3, 11, 4, 9, 9, 1, 9, \ 14, 14, 1, 3, 9, 2, 1, 3, 3, 1, 7, 2, 2, 39}, {3, 6, 4, 2, 1, 6, 2, 6, 1, 1, 7, 7, 17, 10, 9, 8, 2, 7, 6, 1, 8, 6, 1, 2, 6, 4, 3, 6, 10, 28}, {3, 6, 6, 2, 7, 8, 4, 3, 8, 4, 12, 3, 9, 3, 12, 3, 6, \ 6, 2, 7, 8, 4, 3, 8, 4, 12, 3, 9, 3, 12}, {3, 6, 6, 9, 12, 3, 8, 4, 11, 1, \ 3, 9, 2, 1, 12, 3, 6, 6, 9, 12, 3, 8, 4, 11, 1, 3, 9, 2, 1, 12}, {3, 7, 1, 7, 8, 1, 6, 4, 2, 2, 13, 6, 9, 1, 17, 6, 4, 4, 7, 8, 4, 3, 6, 1, 1, 13, 3, 10, 2, 21}, {3, 7, 8, 9, 6, 4, 2, 2, 13, 2, 4, 9, 1, 1, 16, 6, 4, 11, 12, 3, 6, 1, 1, 13, 2, 1, 10, 2, 2, 19}, {3, 7, 8, 11, 4, 4, 2, 7, 6, 8, 1, 6, 3, 6, 13, 4, 6, 9, 12, 3, 4, 3, 6, 6, \ 7, 2, 6, 2, 7, 14}, {3, 8, 2, 9, 2, 2, 6, 7, 2, 13, 6, 3, 10, 2, 7, 8, 2, 9, 13, 2, 7, 6, 2, 2, 9, 2, 6, 2, 13, 15}, {3, 9, 1, 2, 1, 10, 1, 7, 8, 7, \ 14, 1, 9, 2, 1, 10, 4, 12, 3, 12, 6, 1, 8, 1, 3, 3, 7, 8, 11, 15}, {3, 9, 3, 3, 3, 1, 7, 2, 2, 11, 15, 13, 3, 6, 1, 8, 1, 2, 12, 3, 11, 4, 9, 2, 7, 1, 7, 2, 14, 15}, {3, 9, 3, 12, 4, 8, 3, 4, 8, 7, 2, 6, 6, 3, 12, 3, \ 9, 3, 12, 4, 8, 3, 4, 8, 7, 2, 6, 6, 3, 12}, {3, 10, 3, 3, 7, 4, 2, 2, 8, 7, 11, 3, 10, 3, 10, 4, 2, 10, 7, 11, 3, 1, 9, \ 3, 3, 7, 4, 2, 10, 18}, {3, 11, 1, 3, 3, 8, 3, 1, 3, 8, 3, 4, 11, 13, 15, 3, 11, 1, 3, 3, 8, 3, 1, 3, 8, 3, 4, 11, 13, 15}, {3, 12, 6, 1, 2, 7, 2, 4, 2, 13, 2, 9, 12, 6, 9, 1, 2, 12, 7, 2, 9, 4, 2, 12, 1, 2, 7, 2, 12, 15}, {3, 15, 6, 6, 3, 1, 2, 1, 14, 1, 8, 6, 1, 17, 6, 3, 1, 14, 12, 3, 3, 1, 14, 1, 2, 6, 4, 2, 1, 23}, {4, 2, 6, 7, 2, 6, 6, 1, 2, 6, 6, 12, 3, 12, 15, 4, 2, 6, 7, 2, 6, 6, 1, 2, \ 6, 6, 12, 3, 12, 15}, {4, 3, 1, 7, 2, 1, 3, 13, 2, 15, 13, 3, 1, 7, 2, 1, 12, 4, 11, 6, 13, 2, 1, 1, 9, 1, 3, 13, 11, 15}, {4, 3, 4, 3, 1, 3, 3, 1, 8, 3, 4, 11, 16, 7, 3, 7, 9, 7, 8, 3, 4, 8, 3, 1, 3, 4, 3, 4, 3, 39}, {4, 3, 7, 1, 8, 7, 3, 3, 1, 14, 13, 2, 1, 7, 9, 7, 3, 4, 8, 15, 4, 2, 1, 7, 7, 2, 10, 3, 1, 23}, {4, 3, 8, 4, 3, 8, 6, 1, 6, 8, 7, 8, 7, 6, 11, 4, 3, 8, 4, 3, 8, 6, 1, 6, 8, 7, 8, 7, 6, 11}, {4, 4, 2, 9, 2, 1, 1, 8, 3, 3, 1, 14, 8, 10, 11, 10, 3, 4, 11, 3, 1, 7, 4, 3, 1, 2, 11, 1, 9, 29}, {4, 6, 1, 2, 6, 2, 6, 1, 6, 8, 1, 14, 3, 10, 1, 10, 13, 9, 6, 8, 1, 2, 4, 6, 1, 1, 1, 8, 6, 33}, {4, 6, 1, 2, 10, 3, 6, 2, 7, 1, 18, 4, 6, 3, 10, 9, 2, 7, 1, 14, 4, 4, 6, 1, 2, 10, 9, 2, 8, 18}, {4, 6, 8, 1, 3, 4, 2, 6, 2, 1, 15, 8, 10, 8, 8, 2, \ 6, 2, 13, 3, 8, 4, 3, 3, 8, 4, 4, 2, 8, 24}, {4, 6, 9, 2, 1, 9, 3, 3, 1, 14, 1, 7, 8, 2, 11, 10, 3, 15, 3, 8, 4, 3, 1, 2, 11, 1, 1, 8, 7, 22}, {4, 7, 3, 1, 3, 3, 9, 3, 4, 11, 4, 12, 3, 4, 3, 7, \ 9, 15, 3, 12, 3, 1, 3, 4, 3, 4, 3, 1, 15, 23}, {4, 7, 3, 1, 3, 8, 1, 3, 3, 4, 4, 7, 16, 10, 13, 3, 7, 4, 4, 3, 8, 4, 3, 1, \ 7, 3, 4, 9, 10, 23}, {4, 9, 1, 1, 2, 10, 1, 2, 1, 12, 3, 15, 3, 10, 3, 10, \ 3, 13, 2, 13, 2, 1, 3, 9, 1, 2, 1, 10, 4, 29}, {4, 9, 2, 1, 3, 3, 6, 2, 1, 3, 3, 6, 3, 6, 38, 4, 9, 2, 1, 3, 3, 6, 2, 1, 3, 3, 6, 3, 6, 38}, {4, 9, 3, 3, 3, 6, 3, 3, 3, 6, 3, 6, 8, 15, 15, 4, 9, 3, 3, 3, 6, 3, 3, 3, 6, 3, 6, 8, 15, 15}, {4, 10, 1, 2, 1, 3, 9, 1, 1, 1, 14, 1, 16, 10, 7, 9, 1, 14, 2, 1, 12, 2, 1, 1, 10, 3, 1, 3, 10, 29}, {4, 10, 1, 2, 1, 9, 3, 1, 2, 13, 2, 13, 3, 10, 3, 10, 3, 15, 3, 12, 1, 2, 1, 10, 2, 1, 1, 9, 4, 29}, {4, 10, 1, 3, 9, 3, 3, 4, 4, 7, 8, 8, 7, 3, 13, \ 3, 7, 8, 3, 12, 3, 1, 7, 3, 4, 8, 1, 10, 4, 19}, {4, 10, 1, 8, 4, 3, 7, 1, 3, 12, 3, 8, 7, 3, 13, 3, 7, 8, 8, 7, 4, 4, 3, 3, \ 9, 3, 1, 10, 4, 19}, {4, 11, 4, 9, 2, 4, 3, 6, 3, 6, 6, 3, 6, 9, 14, 4, 11, 4, 9, 2, 4, 3, 6, 3, 6, 6, 3, 6, 9, 14}, {4, 14, 1, 8, 6, 1, 2, 6, 6, 3, 6, 3, 6, 9, 15, 4, 14, 1, 8, 6, 1, 2, 6, 6, \ 3, 6, 3, 6, 9, 15}, {6, 1, 2, 4, 2, 1, 1, 7, 4, 2, 9, 4, 23, 1, 9, 1, 13, 9, 4, 2, 9, 4, 2, 6, 1, 2, 4, 3, 1, 43}, {6, 1, 2, 6, 1, 1, 7, 6, 9, 4, 15, 8, 1, 6, 3, 1, 13, 9, 6, 9, 6, 6, 1, 2, 4, 3, 1, 11, 15, 17}, {6, 1, 3, 9, 2, 2, 6, 4, 3, 1, 23, 7, 3, 9, 4, 6, 4, 3, 1, 14, 9, 6, 1, 3, 9, 4, 6, 4, 4, 23}, {6, 1, 14, 1, 2, 12, 1, 2, 4, 11, 4, 2, 13, 2, 15, 6, 1, 14, 1, 2, 12, 1, 2, 4, 11, 4, 2, 13, 2, 15}, {6, 2, 7, 2, 1, 6, 8, 4, 3, 8, 7, 12, 2, 7, 3, 12, 6, 9, 2, 7, 8, 4, 2, 1, 8, 1, 6, 12, 9, 15}, {6, 2, 7, 3, 6, 12, 3, 8, 7, 8, 4, 2, 7, 2, 1, 12, 6, \ 9, 9, 12, 2, 1, 8, 1, 6, 8, 4, 9, 2, 13}, {6, 3, 6, 3, 3, 1, 2, 6, 3, 3, 1, 2, 9, 4, 38, 6, 3, 6, 3, 3, 1, 2, 6, 3, 3, 1, 2, 9, 4, 38}, {6, 3, 8, 2, 2, 7, 4, 4, 2, 9, 13, 9, 6, 4, 9, 2, 6, 2, 9, 4, 11, 4, 3, 8, 2, 9, 2, 8, 7, 15}, {6, 4, 7, 10, 1, 1, 7, 2, 6, 15, 1, 6, 4, 7, 10, 1, 8, \ 2, 21, 1, 6, 4, 4, 3, 10, 1, 1, 7, 2, 22}, {6, 4, 11, 2, 6, 3, 4, 2, 1, 8, \ 6, 7, 2, 8, 19, 7, 3, 12, 2, 7, 2, 4, 2, 2, 7, 6, 6, 3, 7, 21}, {6, 6, 3, 6, 6, 9, 6, 2, 4, 11, 4, 11, 1, 3, 12, 6, 6, 3, 6, 6, 9, 6, 2, 4, \ 11, 4, 11, 1, 3, 12}, {6, 6, 9, 6, 6, 1, 2, 6, 6, 1, 11, 3, 1, 11, 15, 6, 6, 9, 6, 6, 1, 2, 6, 6, 1, 11, 3, 1, 11, 15}, {6, 7, 3, 8, 3, 1, 2, 9, 3, 1, 11, 3, 3, 6, 10, 14, 13, 11, 3, 1, 2, 6, 3, 3, 1, 3, 8, 3, 3, 30}, {6, 7, 8, 7, 8, 6, 1, 6, 8, 3, 4, 8, 3, 4, 11, 6, 7, 8, 7, 8, 6, 1, 6, 8, 3, 4, 8, 3, 4, 11}, {6, 8, 1, 1, 1, 6, 4, 2, 1, 8, \ 6, 9, 13, 10, 1, 10, 3, 14, 1, 8, 6, 1, 6, 2, 6, 2, 1, 6, 4, 33}, {6, 8, 1, 1, 2, 9, 2, 1, 3, 4, 7, 4, 18, 10, 11, 3, 7, 7, 1, 3, 11, 1, 3, 3, 8, 2, 2, 9, 10, 23}, {6, 8, 2, 1, 6, 4, 2, 9, 6, 9, 7, 6, 9, 1, 1, 10, 3, 14, 9, 6, 7, 2, 6, 2, 1, 6, 4, 3, 15, 15}, {6, 9, 1, 2, 9, 3, 3, 4, 7, 4, 11, 7, 8, 2, 11, 3, 7, 8, 3, 12, 3, 3, 8, 2, \ 2, 9, 2, 8, 7, 16}, {6, 10, 8, 8, 1, 6, 4, 4, 7, 6, 2, 4, 9, 1, 14, 3, 10, \ 11, 8, 4, 3, 7, 1, 7, 6, 2, 1, 10, 2, 15}, {7, 1, 2, 7, 4, 15, 2, 1, 12, 3, 6, 7, 1, 1, 1, 7, 21, 13, 9, 6, 1, 1, 1, 1, 7, 4, 3, 14, 1, 21}, {7, 2, 6, 2, 6, 1, 2, 4, 6, 2, 1, 14, 14, 10, 9, 4, 6, 3, 6, 8, 1, 6, 7, 1, \ 1, 8, 6, 3, 10, 24}, {7, 2, 6, 2, 7, 6, 6, 3, 4, 3, 12, 9, 6, 4, 13, 6, 3, \ 6, 1, 8, 6, 7, 2, 4, 4, 11, 8, 7, 3, 14}, {7, 3, 1, 3, 7, 1, 15, 6, 1, 14, \ 2, 10, 1, 2, 1, 7, 16, 7, 8, 8, 7, 3, 1, 2, 1, 7, 7, 15, 1, 16}, {7, 3, 6, 6, 7, 2, 2, 4, 2, 7, 2, 12, 3, 7, 19, 8, 2, 7, 6, 8, 1, 2, 4, 3, 6, 2, 11, 4, 6, 21}, {7, 3, 12, 1, 3, 7, 4, 4, 3, 12, 3, 1, 10, 4, 9, 10, 4, 15, 4, 4, 7, 3, 1, 3, 9, 3, 3, 4, 11, 16}, {7, 4, 4, 2, 9, 2, 1, 9, 3, 3, 15, 8, 8, 2, 11, 2, 8, 3, 4, 11, 3, 8, 4, 3, \ 3, 11, 1, 9, 7, 15}, {7, 7, 2, 6, 2, 7, 2, 4, 6, 3, 14, 14, 1, 9, 6, 3, 4, \ 6, 3, 6, 9, 6, 7, 2, 8, 6, 3, 10, 2, 15}, {7, 8, 1, 1, 11, 2, 1, 3, 4, 8, 3, 15, 3, 10, 11, 2, 8, 7, 1, 14, 1, 3, 3, 9, 1, 2, 9, 6, 4, 22}, {7, 8, 1, 2, 8, 3, 1, 3, 4, 2, 9, 19, 9, 10, 3, 1, 7, 2, 6, 3, 12, 3, 4, 9, \ 2, 8, 3, 8, 2, 21}, {7, 8, 2, 1, 8, 2, 11, 2, 13, 2, 11, 2, 6, 2, 1, 10, 2, 15, 11, 11, 2, 2, 6, 1, 6, 2, 2, 11, 6, 15}, {7, 8, 2, 9, 2, 2, 8, 3, 3, 12, 3, 8, 7, 3, 11, 2, 8, 7, 11, 4, 7, 4, 3, 3, \ 9, 2, 1, 9, 6, 16}, {7, 8, 2, 9, 2, 8, 3, 4, 11, 4, 9, 2, 6, 2, 9, 4, 6, 9, 13, 9, 2, 4, 4, 7, 2, 2, 8, 3, 6, 15}, {8, 1, 3, 3, 4, 3, 1, 7, 6, 2, 13, 15, 3, 3, 7, 3, 8, 8, 7, 8, 7, 6, 2, 1, 3, 7, 2, 1, 14, 24}, {8, 1, 6, 2, 4, 2, 4, 3, 6, 2, 1, 12, 18, 8, 10, 3, 8, 1, 6, 6, 2, 7, 6, 2, \ 1, 8, 4, 6, 12, 21}, {8, 2, 1, 6, 4, 2, 1, 8, 6, 1, 8, 7, 6, 10, 1, 10, 17, 9, 6, 1, 6, 2, 6, 1, 1, 1, 6, 4, 3, 36}, {8, 2, 4, 4, 8, 3, 3, 4, 8, 3, 13, 2, 6, 2, 8, 8, 10, 8, 15, 1, 2, 6, 2, 4, \ 3, 1, 8, 6, 4, 24}, {8, 2, 4, 9, 1, 2, 6, 6, 3, 1, 14, 4, 10, 4, 10, 8, 6, \ 4, 11, 3, 4, 8, 2, 1, 3, 10, 2, 6, 10, 18}, {8, 2, 6, 6, 1, 3, 6, 6, 6, 15, 1, 10, 4, 2, 6, 4, 6, 6, 15, 7, 8, 2, 4, 2, 6, 4, 3, 3, 12, 16}, {8, 2, 11, 2, 1, 7, 7, 1, 3, 12, 3, 3, 10, 2, 9, 10, 7, 15, 1, 6, 8, 1, 1, 2, 9, 3, 3, 4, 11, 18}, {8, 6, 1, 2, 6, 3, 3, 1, 2, 6, 3, 3, 3, 9, 34, 8, 6, 1, 2, 6, 3, 3, 1, 2, 6, 3, 3, 3, 9, 34}, {8, 6, 3, 6, 3, 3, 3, 6, 3, 3, \ 3, 9, 4, 15, 15, 8, 6, 3, 6, 3, 3, 3, 6, 3, 3, 3, 9, 4, 15, 15}, {8, 7, 1, 7, 3, 1, 2, 1, 8, 1, 3, 15, 19, 10, 3, 1, 8, 1, 6, 8, 4, 3, 8, 4, \ 4, 10, 1, 2, 10, 21}, {8, 7, 3, 9, 2, 1, 1, 10, 1, 2, 13, 2, 9, 10, 12, 1, \ 10, 4, 12, 2, 1, 8, 4, 2, 4, 9, 2, 2, 10, 19}, {8, 7, 8, 1, 3, 3, 4, 2, 1, 1, 13, 15, 18, 3, 3, 4, 3, 1, 7, 8, 7, 6, 2, 13, 3, 3, 7, 2, 1, 23}, {9, 1, 3, 4, 2, 9, 3, 1, 11, 3, 1, 13, 1, 8, 1, 9, \ 24, 4, 11, 2, 1, 1, 7, 1, 3, 3, 1, 2, 12, 29}, {9, 1, 3, 10, 1, 2, 1, 12, 2, 1, 14, 1, 3, 10, 3, 10, 16, 15, 2, 1, 3, 9, 1, 1, 1, 1, 10, 3, 1, 33}, {9, 1, 11, 2, 1, 3, 4, 7, 1, 3, 11, 4, 3, 10, 11, 10, 8, 14, 1, 3, 3, 8, 1, 1, 2, 9, 2, 4, 4, 29}, {9, 1, 12, 1, 8, 2, 4, 2, 2, 11, 8, 9, 1, 13, 8, 2, 6, 2, 11, 8, 7, 2, 1, 12, 1, 8, 2, 6, 2, 19}, {9, 3, 3, 3, 6, 2, 1, 3, 3, 6, 2, 1, 6, 8, 34, 9, 3, 3, 3, 6, 2, 1, 3, 3, 6, 2, 1, 6, 8, 34}, {9, 3, 3, 4, 3, 8, 6, 2, 13, 2, 13, 2, 1, 3, 7, 3, 8, 15, 15, 6, 2, 1, 3, 7, 2, 1, 1, 13, 2, 22}, {9, 6, 1, 2, 6, 6, 1, 2, 6, 3, 4, 2, 9, 15, 18, 9, 6, 1, 2, 6, 6, 1, 2, 6, 3, 4, 2, 9, 15, 18}, {9, 6, 1, 3, 7, 1, 1, 2, 12, 1, 14, 1, 11, 7, 3, 7, 4, 15, 12, 1, 2, 9, 3, 1, 3, 3, 7, 1, 1, 32}, {9, 6, 2, 2, 9, 2, 2, 4, 2, 9, 4, 15, 3, 10, 9, 2, 8, 7, 2, 13, 2, 4, 3, 8, \ 2, 2, 7, 8, 2, 22}, {9, 6, 2, 4, 6, 3, 6, 2, 1, 12, 2, 15, 1, 8, 10, 3, 9, \ 6, 6, 9, 6, 2, 1, 8, 4, 2, 4, 11, 1, 21}, {9, 6, 3, 6, 3, 6, 6, 2, 1, 6, 8, 1, 14, 4, 15, 9, 6, 3, 6, 3, 6, 6, 2, 1, 6, 8, 1, 14, 4, 15}, {9, 6, 3, 6, 6, 3, 6, 3, 4, 2, 9, 4, 11, 4, 14, 9, 6, 3, 6, 6, 3, 6, 3, 4, 2, 9, 4, 11, 4, 14}, {9, 6, 7, 1, 1, 2, 4, 2, 1, 6, 2, 15, 26, 1, 7, 2, 1, \ 6, 6, 9, 2, 4, 9, 2, 11, 1, 3, 6, 1, 27}, {9, 12, 6, 1, 8, 1, 2, 4, 8, 7, 2, 9, 6, 12, 3, 7, 2, 12, 7, 8, 3, 4, 8, 6, 1, 2, 7, 2, 6, 15}, {10, 2, 1, 10, 4, 4, 8, 3, 4, 8, 6, 1, 8, 1, 3, 10, 19, 15, 3, 1, 8, 1, 2, 1, 3, 7, 1, 7, 8, 21}, {10, 2, 2, 9, 4, 2, 4, 8, 1, 2, 12, 4, 10, 1, 12, 10, 9, 2, 13, 2, 1, 10, 1, 1, 2, 9, 3, 7, 8, 19}, {10, 2, 4, 7, 3, 3, 9, 1, 3, 11, 7, 10, 2, 4, 10, 3, 10, 3, 11, 7, 8, 2, 2, \ 4, 7, 3, 3, 10, 3, 18}, {10, 3, 1, 3, 10, 1, 1, 2, 12, 1, 2, 14, 1, 9, 7, 10, 16, 1, 14, 1, 1, 1, 9, 3, 1, 2, 1, 10, 4, 29}, {10, 3, 4, 9, 1, 1, 13, 2, 1, 12, 3, 1, 10, 4, 3, 10, 16, 13, 2, 2, 10, 1, 2, 1, 3, 9, 1, 2, 15, 16}, {10, 3, 4, 10, 1, 3, 12, 1, 2, 13, 1, 1, 9, 4, 3, 10, 16, 15, 2, 1, 9, 3, 1, 2, 1, 10, 2, 2, 13, 16}, {10, 3, 6, 8, 1, 1, 7, 6, 1, 8, 6, 3, 6, 4, 9, 10, 14, 14, 1, 2, 6, 4, 2, 1, 6, 2, 6, 2, 7, 24}, {10, 4, 9, 6, 4, 8, 1, 2, 12, 1, 3, 10, 1, 1, 11, 10, 11, 15, 1, 10, 1, 1, 2, 9, 3, 1, 6, 8, 1, 18}, {10, 6, 2, 10, 3, 1, 2, 8, 4, 3, 11, 4, 6, 8, 10, 4, 10, 4, 14, 1, 3, 6, 6, \ 2, 1, 9, 4, 2, 8, 18}, {10, 6, 3, 4, 6, 2, 1, 6, 8, 1, 6, 7, 2, 8, 9, 10, 17, 7, 7, 1, 1, 6, 2, 6, 1, 2, 4, 6, 3, 28}, {10, 9, 2, 2, 8, 3, 3, 1, 11, 3, 1, 7, 7, 3, 11, 10, 18, 4, 7, 4, 3, 1, 2, 9, 2, 1, 1, 8, 6, 23}, {10, 9, 4, 3, 7, 1, 3, 4, 8, 3, 4, 4, 7, 3, 13, 10, \ 16, 7, 4, 4, 3, 3, 1, 8, 3, 1, 3, 7, 4, 23}, {10, 13, 6, 3, 4, 2, 1, 8, 4, 2, 7, 2, 4, 4, 19, 10, 14, 7, 2, 4, 2, 2, 7, 4, 2, 6, 3, 4, 3, 21}, {11, 1, 3, 7, 4, 3, 1, 8, 1, 2, 12, 15, 4, 10, 7, 1, 9, 2, 4, 11, 4, 8, 3, 1, 10, 1, 6, 9, 1, 21}, {11, 1, 3, 11, 1, 6, 6, 2, 1, 6, 6, 9, 6, 6, 15, 11, 1, 3, 11, 1, 6, 6, 2, 1, 6, 6, 9, 6, 6, 15}, {11, 1, 10, 4, 3, 9, 1, 2, 12, 7, 8, 4, 3, 7, 7, 1, \ 9, 2, 15, 12, 3, 1, 10, 1, 6, 1, 8, 1, 6, 15}, {11, 2, 2, 7, 1, 3, 3, 3, 9, 3, 15, 14, 2, 7, 1, 7, 2, 9, 4, 11, 3, 12, 2, 1, 8, 1, 6, 3, 13, 15}, {11, 2, 8, 1, 2, 4, 4, 7, 4, 11, 6, 11, 4, 6, 1, 8, 2, 11, 11, 4, 11, 2, 2, 8, 1, 2, 6, 2, 13, 15}, {11, 4, 3, 3, 9, 2, 1, 1, 8, 6, 1, 15, 7, 10, 9, 2, 10, 3, 3, 12, 3, 1, 7, 7, 1, 2, 11, 2, 8, 18}, {11, 4, 3, 3, 9, 3, 1, 3, 7, 4, 4, 15, 4, 10, 9, 4, 10, 1, 3, 12, 3, 4, 4, 7, 3, 1, 12, 3, 7, 16}, {11, 8, 2, 7, 2, 4, 3, 1, 11, 3, 8, 7, 4, 10, 7, 2, 8, 11, 11, 4, 3, 4, 8, 2, 1, 6, 2, 7, 1, 22}, {11, 8, 7, 3, 3, 1, 8, 1, 6, 12, 3, 12, 4, 10, 1, 2, 9, 1, 14, 7, 8, 7, 1, 10, 1, 2, 1, 9, 3, 15}, {12, 2, 1, 3, 3, 1, 7, 1, 1, 2, 11, 4, 24, 9, 1, 8, 1, 13, 1, 3, 11, 1, 3, 9, 2, 4, 3, 1, 9, 29}, {12, 2, 7, 2, 1, 12, 2, 4, 9, 2, 7, 12, 2, 1, 9, 6, \ 12, 9, 2, 13, 2, 4, 2, 7, 2, 1, 6, 12, 3, 15}, {12, 3, 3, 3, 1, 8, 1, 2, 11, 4, 11, 13, 2, 7, 1, 8, 1, 14, 3, 12, 3, 9, 2, \ 4, 3, 1, 7, 2, 13, 16}, {12, 3, 3, 4, 6, 2, 4, 2, 8, 7, 15, 6, 6, 4, 6, 2, \ 4, 10, 1, 15, 6, 6, 6, 3, 1, 6, 6, 2, 8, 16}, {12, 3, 6, 3, 6, 4, 2, 6, 1, 8, 7, 14, 1, 11, 6, 4, 8, 3, 6, 9, 6, 6, 1, 8, \ 3, 4, 6, 8, 1, 17}, {12, 3, 12, 6, 6, 2, 1, 6, 6, 2, 7, 6, 2, 4, 15, 12, 3, 12, 6, 6, 2, 1, 6, 6, 2, 7, 6, 2, 4, 15}, {12, 4, 6, 1, 3, 3, 1, 3, 11, 15, 1, 12, 4, 6, 1, 3, 4, 3, 26, 1, 12, 2, 2, \ 6, 1, 3, 3, 1, 3, 27}, {12, 6, 4, 8, 1, 2, 6, 7, 2, 6, 6, 1, 8, 3, 10, 8, 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7, 3, 3, 15, 13, 2, 6, 7, 1, 3, 4, 3, 3, 1, 8, 24}, {14, 3, 7, 6, 2, 1, 3, 11, 1, 3, 9, 3, 3, 8, 10, 6, 17, 13, 2, 1, 3, 8, 3, 1, 3, 3, 6, 3, 3, 24}, {15, 1, 3, 4, 3, 4, 3, 1, 3, 12, 3, 15, 9, 7, 3, 4, \ 3, 12, 4, 11, 4, 3, 9, 3, 3, 1, 3, 7, 4, 23}, {15, 1, 10, 1, 2, 1, 8, 1, 3, 11, 4, 11, 8, 10, 3, 1, 9, 6, 12, 3, 8, 4, 4, \ 7, 3, 1, 2, 9, 1, 21}, {15, 2, 1, 8, 1, 3, 3, 1, 2, 12, 1, 15, 13, 9, 1, 3, 6, 9, 3, 12, 3, 1, 13, 1, 1, 7, 1, 7, 2, 24}, {15, 2, 2, 8, 1, 2, 4, 2, 2, 11, 2, 15, 11, 10, 1, 2, 8, 7, 4, 11, 4, 2, 11, 2, 2, 6, 1, 8, 2, 22}, {15, 3, 1, 11, 3, 1, 2, 1, 11, 1, 2, 1, 14, 1, 23, 15, 3, 1, 11, 3, 1, 2, 1, 11, 1, 2, 1, 14, 1, 23}, {15, 3, 4, 2, 4, 2, 2, 2, 2, 15, 15, 12, 4, 2, 4, 2, 2, 2, 11, 15, 6, 12, 3, 1, 2, 4, 4, 2, 2, 24}, {15, 9, 2, 4, 3, 6, 2, 1, 6, 6, 2, 1, 6, 9, 18, 15, 9, 2, 4, 3, 6, 2, 1, 6, 6, 2, 1, 6, 9, 18}, {15, 9, 3, 3, 4, 2, 1, 1, 13, 2, 13, 2, 16, 3, 3, 4, 3, 8, 15, 6, 2, 13, 2, \ 1, 3, 7, 2, 1, 1, 22}, {15, 11, 1, 3, 4, 2, 1, 6, 6, 9, 6, 9, 13, 1, 3, 6, \ 1, 8, 15, 4, 9, 6, 7, 1, 1, 6, 2, 1, 6, 17}, {15, 12, 1, 1, 1, 6, 1, 1, 6, 7, 8, 7, 21, 1, 2, 7, 1, 7, 14, 1, 6, 8, 7, 6, 1, 1, 7, 1, 1, 22}, {16, 3, 7, 4, 2, 1, 1, 8, 1, 6, 11, 16, 10, 4, 2, 1, 9, 1, 6, 11, 4, 12, 3, 7, 4, 2, 1, 9, 1, 17}}\ \>", "Output"] }, Closed]], Cell[BoxData[ RowBox[{ RowBox[{"This", " ", "is", " ", StyleBox["known", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], "to", " ", "be", " ", "incomplete"}], StyleBox[","], StyleBox[" "], RowBox[{"as", " ", "some", " ", "elements", " ", "of", " ", StyleBox[\(r\_\(\(120\)\(\ \)\)\), FontWeight->"Plain"], "are", " ", \(\(missing\)\(.\)\)}]}]], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["108", "Subsection"], Cell[TextData[{ "Completing any element of the one orbit that forms ", Cell[BoxData[ \(TraditionalForm\`S\_108\)]], " gives 672 solutions, 252 of which are acyclic. Here are the 72 new ones \ (12 orbits of 6 each) among them (no palindroms):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(Complement[%, \ R108]\)\)\)], "Input"], Cell[BoxData[ \({{1, 1, 4, 4, 9, 1, 16, 2, 4, 13, 1, 8, 8, 1, 1, 4, 14, 16}, {1, 1, 4, 5, 8, 1, 16, 1, 5, 13, 1, 9, 7, 1, 1, 4, 13, 17}, {1, 1, 5, 4, 7, 2, 16, 1, 6, 11, 2, 9, 7, 1, 1, 5, 11, 18}, {1, 1, 6, 2, 8, 1, 17, 2, 6, 10, 1, 9, 8, 1, 1, 6, 10, 18}, {1, 4, 1, 8, 5, 4, 13, 1, 5, 13, 4, 9, 4, 1, 4, 1, 13, 17}, {1, 4, 2, 7, 4, 5, 13, 1, 6, 11, 5, 9, 4, 1, 4, 2, 11, 18}, {1, 4, 5, 1, 7, 1, 17, 5, 6, 7, 1, 9, 8, 1, 4, 6, 7, 18}, {1, 5, 2, 9, 2, 7, 10, 1, 5, 13, 7, 9, 1, 1, 5, 2, 11, 17}, {1, 5, 5, 8, 1, 9, 7, 1, 5, 13, 10, 7, 1, 1, 4, 5, 8, 17}, {1, 5, 8, 5, 4, 9, 4, 1, 5, 13, 13, 4, 1, 4, 1, 8, 5, 17}, {1, 5, 11, 2, 7, 9, 1, 1, 5, 13, 16, 1, 1, 5, 2, 9, 2, 17}, {1, 6, 1, 9, 1, 8, 10, 1, 6, 11, 8, 9, 1, 1, 6, 1, 10, 18}, {1, 6, 4, 7, 2, 9, 7, 1, 6, 11, 11, 7, 1, 1, 5, 4, 7, 18}, {1, 6, 7, 4, 5, 9, 4, 1, 6, 11, 14, 4, 1, 4, 2, 7, 4, 18}, {1, 6, 10, 1, 8, 9, 1, 1, 6, 11, 17, 1, 1, 6, 1, 9, 1, 18}, {1, 7, 2, 4, 4, 1, 17, 8, 6, 4, 1, 9, 8, 1, 7, 6, 4, 18}, {1, 9, 1, 6, 1, 1, 17, 11, 6, 1, 1, 9, 8, 1, 10, 6, 1, 18}, {1, 9, 4, 4, 1, 1, 16, 14, 4, 1, 1, 8, 8, 1, 13, 4, 2, 16}, {2, 2, 2, 7, 7, 2, 14, 2, 4, 14, 2, 9, 5, 2, 2, 2, 14, 16}, {2, 2, 4, 5, 5, 4, 14, 2, 6, 10, 4, 9, 5, 2, 2, 4, 10, 18}, {2, 2, 6, 1, 7, 2, 16, 4, 6, 8, 2, 9, 7, 2, 2, 6, 8, 18}, {2, 4, 1, 9, 4, 5, 11, 2, 4, 14, 5, 9, 2, 2, 4, 1, 13, 16}, {2, 4, 4, 9, 1, 8, 8, 2, 4, 14, 8, 8, 1, 1, 4, 4, 10, 16}, {2, 4, 7, 7, 2, 9, 5, 2, 4, 14, 11, 5, 2, 2, 2, 7, 7, 16}, {2, 4, 10, 4, 5, 9, 2, 2, 4, 14, 14, 2, 2, 4, 1, 9, 4, 16}, {2, 5, 1, 8, 2, 7, 11, 2, 6, 10, 7, 9, 2, 2, 5, 1, 10, 18}, {2, 5, 4, 2, 5, 2, 16, 7, 6, 5, 2, 9, 7, 2, 5, 6, 5, 18}, {2, 6, 2, 8, 1, 9, 8, 2, 6, 10, 10, 8, 1, 1, 6, 2, 8, 18}, {2, 6, 5, 5, 4, 9, 5, 2, 6, 10, 13, 5, 2, 2, 4, 5, 5, 18}, {2, 6, 8, 2, 7, 9, 2, 2, 6, 10, 16, 2, 2, 5, 1, 8, 2, 18}, {2, 8, 1, 5, 2, 2, 16, 10, 6, 2, 2, 9, 7, 2, 8, 6, 2, 18}, {2, 9, 2, 5, 1, 1, 16, 13, 5, 1, 1, 9, 7, 2, 11, 5, 1, 17}, {4, 1, 5, 4, 4, 5, 13, 4, 6, 8, 5, 9, 4, 4, 1, 5, 8, 18}, {4, 1, 6, 2, 5, 4, 14, 5, 6, 7, 4, 9, 5, 4, 1, 6, 7, 18}, {4, 4, 2, 7, 1, 8, 10, 4, 6, 8, 8, 9, 1, 4, 4, 2, 8, 18}, {4, 4, 5, 1, 4, 4, 14, 8, 6, 4, 4, 9, 5, 4, 4, 6, 4, 18}, {4, 6, 1, 7, 2, 9, 7, 4, 6, 8, 11, 7, 2, 2, 6, 1, 7, 18}, {4, 6, 4, 4, 5, 9, 4, 4, 6, 8, 14, 4, 4, 1, 5, 4, 4, 18}, {4, 6, 7, 1, 8, 9, 1, 4, 6, 8, 17, 1, 4, 4, 2, 7, 1, 18}, {4, 7, 2, 4, 1, 4, 14, 11, 6, 1, 4, 9, 5, 4, 7, 6, 1, 18}, {4, 9, 1, 4, 2, 2, 14, 14, 4, 2, 2, 9, 5, 4, 10, 4, 2, 16}, {5, 2, 4, 5, 2, 7, 11, 5, 6, 7, 7, 9, 2, 5, 2, 4, 7, 18}, {5, 2, 6, 1, 4, 5, 13, 7, 6, 5, 5, 9, 4, 5, 2, 6, 5, 18}, {5, 5, 1, 7, 1, 9, 8, 5, 6, 7, 10, 8, 1, 4, 5, 1, 7, 18}, {5, 5, 4, 2, 2, 5, 13, 10, 6, 2, 5, 9, 4, 5, 5, 6, 2, 18}, {5, 6, 2, 5, 4, 9, 5, 5, 6, 7, 13, 5, 4, 1, 6, 2, 5, 18}, {5, 6, 5, 2, 7, 9, 2, 5, 6, 7, 16, 2, 5, 2, 4, 5, 2, 18}, {5, 8, 1, 4, 1, 4, 13, 13, 5, 1, 4, 9, 4, 5, 8, 5, 1, 17}, {7, 1, 5, 4, 1, 8, 10, 7, 6, 5, 8, 9, 1, 7, 1, 5, 5, 18}, {7, 1, 6, 2, 2, 7, 11, 8, 6, 4, 7, 9, 2, 7, 1, 6, 4, 18}, {7, 4, 2, 5, 2, 9, 7, 7, 6, 5, 11, 7, 2, 5, 4, 2, 5, 18}, {7, 4, 5, 1, 1, 7, 11, 11, 6, 1, 7, 9, 2, 7, 4, 6, 1, 18}, {7, 6, 1, 4, 5, 9, 4, 7, 6, 5, 14, 4, 5, 2, 6, 1, 4, 18}, {7, 6, 4, 1, 8, 9, 1, 7, 6, 5, 17, 1, 7, 1, 5, 4, 1, 18}, {7, 7, 2, 2, 2, 5, 11, 14, 4, 2, 5, 9, 2, 7, 7, 4, 2, 16}, {8, 2, 4, 4, 1, 9, 8, 8, 6, 4, 10, 8, 1, 7, 2, 4, 4, 18}, {8, 2, 6, 1, 1, 8, 10, 10, 6, 2, 8, 9, 1, 8, 2, 6, 2, 18}, {8, 5, 1, 4, 4, 9, 5, 8, 6, 4, 13, 5, 4, 4, 5, 1, 4, 18}, {8, 5, 4, 1, 1, 7, 10, 13, 5, 1, 7, 9, 1, 8, 5, 5, 1, 17}, {8, 6, 2, 2, 7, 9, 2, 8, 6, 4, 16, 2, 7, 1, 6, 2, 2, 18}, {10, 1, 5, 2, 2, 9, 7, 10, 6, 2, 11, 7, 2, 8, 1, 5, 2, 18}, {10, 1, 6, 1, 1, 9, 8, 11, 6, 1, 10, 8, 1, 9, 1, 6, 1, 18}, {10, 4, 2, 2, 5, 9, 4, 10, 6, 2, 14, 4, 5, 5, 4, 2, 2, 18}, {10, 4, 4, 1, 1, 8, 8, 14, 4, 2, 8, 8, 1, 9, 4, 4, 2, 16}, {10, 6, 1, 1, 8, 9, 1, 10, 6, 2, 17, 1, 8, 2, 6, 1, 1, 18}, {11, 2, 4, 1, 4, 9, 5, 11, 6, 1, 13, 5, 4, 7, 2, 4, 1, 18}, {11, 2, 5, 1, 1, 9, 7, 13, 5, 1, 10, 7, 2, 9, 2, 5, 1, 17}, {11, 5, 1, 1, 7, 9, 2, 11, 6, 1, 16, 2, 7, 4, 5, 1, 1, 18}, {13, 1, 4, 1, 4, 9, 4, 13, 5, 1, 13, 4, 5, 8, 1, 4, 1, 17}, {13, 1, 4, 2, 2, 9, 5, 14, 4, 2, 11, 5, 4, 9, 1, 4, 2, 16}, {13, 4, 1, 1, 7, 9, 1, 13, 5, 1, 16, 1, 8, 5, 4, 1, 1, 17}, {14, 2, 2, 2, 5, 9, 2, 14, 4, 2, 14, 2, 7, 7, 2, 2, 2, 16}}\)], "Output", GeneratedCell->False, CellAutoOverwrite->False] }, Closed]], Cell[TextData[{ "Now, I tried completing one of the 672-252 ", StyleBox["periodic", FontWeight->"Bold", FontSlant->"Italic"], " complements, for instance {1,8,1,7,1,1,7,9,19,1,8,1,7,1,1,7,9,19}, This \ gives some 48 ", StyleBox["aperiodic", FontWeight->"Bold", FontSlant->"Italic"], " solutions. Strangely not all of them are complements of an acyclic tile, \ indeed we find back only the known 3 elements of ", Cell[BoxData[ \(TraditionalForm\`S\_108\)]], " :" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(%\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"someAcyclicTiles", " ", "=", RowBox[{"{", RowBox[{\({3, 3, 21, 3, 3, 75}\), ",", \({3, 12, 12, 3, 12, 66}\), ",", \({3, 21, 3, 3, 21, 57}\), ",", \({3, 24, 3, 3, 27, 48}\), ",", \({3, 24, 3, 12, 27, 39}\), ",", \({3, 24, 3, 21, 27, 30}\), ",", \({3, 27, 3, 27, 21, 27}\), ",", \({3, 27, 12, 27, 12, 27}\), ",", \({6, 6, 15, 6, 6, 69}\), ",", \({6, 15, 6, 6, 15, 60}\), ",", \({6, 21, 3, 3, 24, 51}\), ",", \({6, 21, 6, 6, 27, 42}\), ",", \({6, 21, 6, 15, 27, 33}\), ",", \({6, 24, 3, 24, 24, 27}\), ",", \({6, 27, 6, 27, 15, 27}\), ",", \({12, 3, 12, 12, 3, 66}\), ",", \({12, 12, 3, 12, 12, 57}\), ",", \({12, 15, 6, 6, 21, 48}\), ",", \({12, 15, 12, 3, 27, 39}\), ",", \({12, 15, 12, 12, 27, 30}\), ",", \({12, 21, 6, 21, 21, 27}\), ",", \({15, 6, 6, 15, 6, 60}\), ",", \({15, 12, 3, 12, 15, 51}\), ",", \({15, 12, 12, 3, 24, 42}\), ",", \({15, 12, 15, 6, 27, 33}\), ",", \({15, 15, 12, 15, 24, 27}\), ",", \({15, 24, 3, 24, 15, 27}\), ",", \({21, 3, 3, 21, 3, 57}\), ",", \({21, 6, 6, 15, 12, 48}\), ",", \({21, 6, 15, 6, 21, 39}\), ",", \({21, 6, 21, 3, 27, 30}\), ",", StyleBox[\({21, 12, 15, 12, 21, 27}\), FontWeight->"Plain"], ",", \({21, 21, 6, 21, 12, 27}\), ",", \({24, 3, 3, 21, 6, 51}\), ",", \({24, 3, 12, 12, 15, 42}\), ",", \({24, 3, 21, 3, 24, 33}\), ",", \({24, 6, 21, 6, 24, 27}\), ",", \({24, 15, 12, 15, 15, 27}\), ",", \({24, 24, 3, 24, 6, 27}\), ",", \({27, 3, 3, 24, 3, 48}\), ",", \({27, 3, 12, 15, 12, 39}\), ",", \({27, 3, 21, 6, 21, 30}\), ",", \({27, 6, 6, 21, 6, 42}\), ",", \({27, 6, 15, 12, 15, 33}\), ",", \({27, 12, 3, 24, 3, 39}\), ",", \({27, 12, 12, 15, 12, 30}\), ",", \({27, 15, 6, 21, 6, 33}\), ",", \({27, 21, 3, 24, 3, 30}\)}], "}"}]}], ";"}]], "Output", GeneratedCell->False, CellAutoOverwrite->False] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Select[someAcyclicTiles, \ canonQ[#, \ \[IndentingNewLine]{1, 6, 4, 7, 2, 9, 7, 1, 6, 11, 11, 7, 1, 1, 5, 4, 7, 18}\[IndentingNewLine]] &]\)], "Input"], Cell[BoxData[ \({{12, 12, 3, 12, 12, 57}, {21, 12, 15, 12, 21, 27}, {24, 3, 21, 3, 24, 33}}\)], "Output"] }, Open ]], Cell[TextData[{ "This puts to light two interesting (or maybe depressing) facts :\n\ \[Bullet] by double complementing of a periodic (not acyclic) tile one may \ sometimes find acyclic canons,\n\[Bullet] some tile may have as complements a \ subset of the complements of some other (more regular ?) tile. For instance, \ the complements of {1,7,10,1,7,10,\[Ellipsis],1,7,10} are 1296 (!) but they \ include the aforementioned ", StyleBox["someAcyclicTiles", "Input"], " which did include already ", Cell[BoxData[ \(TraditionalForm\`S\_108\)]], "." }], "Text"] }, Closed]] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for Macintosh", ScreenRectangle->{{0, 1440}, {0, 818}}, WindowSize->{638, 452}, WindowMargins->{{69, Automatic}, {Automatic, 127}}, StyleDefinitions -> "NaturalColor.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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