

Three circles can form
seven nonoverlapping arcotriangles. 



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
nonoverlapping arcotriangles. An arcotriangle is a
shape formed with three arcs which are crossing in three
vertices of the arcotriangle.
Using four and five circles we can form shapes divided
into 11 and 16 nonoverlapping arcotriangles,
respectively. Can you find those shapes? Note that
circles can be of any necessary sizes. Also, within a
shape you can have some other, nontriangular 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, nontriangular space(s)? This
means that the shape, in fact, should consist just of
nonoverlapping arcotriangles 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. 
