- 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?)
For the count: We use the fact that the Euler characteristic of the sphere is $2$. A generic configuration of great circles will have $ 2\times \frac {n(n-1)}2=n(n-1)$ vertices and $n\times 2(n-1)$ edges. Thus we have $$2=\#F_n -2n^2+2n+n^2-n=\#F_n-n^2+n\implies \boxed {\#F_n=n^2-n+2}$$
As a sanity check we remark that $\#F_1=2,\#F_2=4,\#F_3=9-3+2=8$ as desired.