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 Arco-Triangles*
 Three circles can form seven non-overlapping arco-triangles.
 If you take three circles and overlap them as shown in the diagram at right, you can see that the whole shape is divided into seven non-overlapping arco-triangles. An arco-triangle is a shape formed with three arcs which are crossing in three vertices of the arco-triangle. Using four and five circles we can form shapes divided into 11 and 16 non-overlapping arco-triangles, respectively. Can you find those shapes? Note that circles can be of any necessary sizes. Also, within a shape you can have some other, non-triangular shapes which can be ignored. Some questions. 1. Can you improve the above results? 2. What are the maximal results for six and more circles? 3. What are the maximal results if each resulting shape does not contain free, non-triangular space(s)? This means that the shape, in fact, should consist just of non-overlapping arco-triangles like that formed of three circles; see above right diagram. Write us if you can improve the above results or will find maximal solutions for six and more circles.

 *) This puzzle was inspired by the Kobon Triangle puzzle posed by a famous Japanese math teacher, puzzle writer and creator Kobon Fujimura. The Kobon Triangle puzzle (or Fujimura's Triangle Puzzle) was presented in his The Tokyo Puzzles book published in 1978 by Charles Scribner's Sons with an Introduction by Martin Gardner. ----- Also you can read a big article by Ed Pegg Jr.on Kobon Triangles at the MAA website.

Last Updated: October 14, 2008
Posted: February 9, 2006