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Solutions of the Loculus of Archimedes

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The 536 distinct solutions for the Loculus of Archimedes

Originator: Bill Cutler, Ed Pegg Jr.

More than 2000 years ago, Archimedes invented a dissection game called the Loculus, making it the world's oldest puzzle. In 2003, Bill Cutler enumerated all solutions with the aid of a computer program. Without rotations and reflections, there are 536 distinct solutions.

Examples

Basic Examples

First solution:

In[1]:=
ResourceData["Solutions of the Loculus of Archimedes"][1]
Out[1]=

Visualize the solution:

In[2]:=
Graphics[{GrayLevel[0.9], Table[{Line[{{n, 0}, {n, 12}}], Line[{{0, n}, {12, n}}]}, {n, 0, 12}], GrayLevel[0], ({Line[#1], (Flatten[{With[{lengths = Sort[Abs[Flatten[Apply[Subtract, Partition[#1, 2, 1], {1}]]]], pos = (1/3)*Apply[Plus, Transpose[Which[Length[#1] == 4, Take[#1, 3], Length[#1] == 6, #1[[{1, 4, 2}]], Length[#1] == 5, (3/2)*#1[[{1, 3}]]]], {1}]}, Which @@ Flatten[MapThread[
                              {lengths === #1, {Text[
                   Style[#3, 30, FontFamily -> "Helvetica"], pos], #2}} & , {{{0, 0, 2, 2, 3, 3}, {0, 0, 3, 3, 6, 6}, {0, 1, 1, 2, 4, 6},
                                  {0, 1, 2, 3, 4, 4}, {0, 2, 2, 2, 4, 6}, {0, 2, 2, 2, 10, 12}, {0, 2, 4, 4, 4, 6}, {1, 2, 3, 4, 4, 6}, {2, 2, 2, 4, 8, 10}, {0, 0, 2, 3, 3, 4, 6, 6}, {0, 1, 2, 2, 2, 3, 6, 6}, {0, 0, 0, 1, 2, 2, 2, 3, 6, 6}}, {Red, Green, Blue, Pink, Cyan, Brown, Purple, Yellow, Magenta, Orange, LightRed, LightBlue}, Range[12]}], 1]], Opacity[0.3], Polygon[#1]}] & ) /@ #1} & )[
   ResourceData["Solutions of the Loculus of Archimedes"][1]]}]
Out[2]=

Enrique Zeleny, "Solutions of the Loculus of Archimedes" from the Wolfram Data Repository (2020) 

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