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Geometric Puzzles in the Classroom

# Diarcs and Triarcs

by Michael Keller

This article is reprinted, with permission, from Michael Keller's games and puzzles 'zine (#9, Dec 1989).

For a little background and context, see
Geometric Puzzles in the Classroom: Polyarcs.

The 22 triarcs are shown above. They can be combined with the diarcs to form a 5x8 rectangle with straight sides, or a rectangle with edges alternately convex and concave.

Is a 3x11 rectangle possible with the triarcs alone? What about a 4x10 with diarcs and triarcs?

I also counted the tetrarcs in two ways: by adding either monarc to each triarc in every possible way (eliminating duplicates), and by considering every way to convert each tetratan into a tetrarc (changing exposed hypoteneuses into convex or concave arcs). Both counts added up to 93. There are 6 tetrarcs made up of 4 thin arcs, 22 of 3 thin and 1 thick, 37 of 2 and 2, 22 of 3 thick and 1 thin, and 6 of 4 thick. These figures cover an area of 186 squares, and conceivably might form a 6x31 rectangle alone. Combining them with the lower order sets gives rise to many more possibilities.

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