- What is the maximal number of disjoint regions obtained on the sphere by dividing it with $n$ great circles?
For $n= 1$ we have $2$ regions, for $n=2 $ we have $2^2$, for $n=3$ the number is $2^3$,... what next? - what would be the best approach to this problem? How to use geometrical constraints solving it?
... and ...
- Is it possible to find always $n$ great circles (in the case when the number of regions is maximal) for such division that areas of regions would be equal ? (I suppose not but how to prove it?)