(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 417384, 5791]*) (*NotebookOutlinePosition[ 418029, 5813]*) (* CellTagsIndexPosition[ 417985, 5809]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The Loculus of Archimedes", "Subtitle"], Cell["\<\ A puzzle invented by Archimedes data provided by Bill Cutler, of Bill Cutler Puzzles \ (http://billcutlerpuzzles.com/) code by Ed Pegg Jr.\ \>", "Text"], Cell[CellGroupData[{ Cell["Loculus Data (initialize)", "Section"], Cell[BoxData[ \(\(Loculus\ = \ {{{{0, 0}, {0, 12}, {2, 10}, {0, 0}}, {{0, 0}, {2, 4}, {3, 0}, {0, 0}}, {{0, 0}, {2, 10}, {4, 8}, {0, 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ImageRangeCache->{{{0, 287}, {287, 0}} -> {-0.300063, -0.300063, 0.0439029, \ 0.0439029}}] }, Open ]], Cell["The following code will show all 536 solutions.", "Text"], Cell[BoxData[ \(\(Show[ GraphicsArray[ Partition[ Table[Graphics[Map[Line[#] &, Loculus[\([g]\)]], AspectRatio \[Rule] \ Automatic], {g, 1, 536}], 8]]];\)\)], "Input"], Cell["\<\ Which grid points are most and least likely to have a vertex of one of the \ pieces?\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ranker\ = \ Sort[Map[{Length[#], First[#]} &, Split[Sort[Partition[Flatten[Loculus], 2]]]]]\)], "Input"], Cell[BoxData[ \({{8, {6, 3}}, {12, {4, 11}}, {14, {1, 4}}, {14, {5, 4}}, {14, {5, 8}}, {16, {12, 3}}, {17, {7, 4}}, {19, {11, 8}}, {21, {1, 8}}, {21, {11, 4}}, {28, {7, 8}}, {30, {2, 11}}, {40, {8, 12}}, {41, {4, 10}}, {43, {10, 10}}, {46, {11, 2}}, {51, {1, 2}}, {51, {1, 10}}, {51, {5, 2}}, {51, {7, 10}}, {53, {10, 2}}, {75, {4, 2}}, {80, {7, 2}}, {87, {8, 2}}, {92, {2, 2}}, {110, {0, 8}}, {112, {5, 10}}, {120, {6, 9}}, {133, {12, 8}}, {144, {0, 4}}, {159, {11, 10}}, {221, {6, 8}}, {228, {2, 8}}, {257, {8, 8}}, {284, {12, 4}}, {309, {9, 12}}, {324, {3, 12}}, {399, {6, 4}}, {480, {4, 4}}, {499, {10, 4}}, {500, {9, 0}}, {515, {2, 4}}, {553, {8, 4}}, {669, {3, 0}}, {740, {0, 6}}, {911, {8, 10}}, {1052, {10, 8}}, {1068, {2, 10}}, {1237, {4, 8}}, {1245, {12, 0}}, {1386, {12, 6}}, {1594, {0, 12}}, {1621, {12, 12}}, {1888, {6, 6}}, {1945, {3, 6}}, {1995, {9, 6}}, {2514, {0, 0}}, {2759, {6, 12}}, {3214, {6, 0}}}\)], "Output"] }, Open ]], Cell["Which solutions contain rare vertices?", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Complement[Range[536], Union[Map[First, Position[Loculus, {9, 6}]]]]\)], "Input"], Cell[BoxData[ \({2, 3, 4, 5, 6, 7, 8, 9, 19, 20, 21, 22, 30, 31, 32, 33, 34, 35, 36, 37}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Complement[Range[536], Union[Map[First, Position[Loculus, {6, 6}]]]]\)], "Input"], Cell[BoxData[ \({17, 18, 23, 24}\)], "Output"] }, Open 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