$N$ circles in a plane all intersect each other such that every circle intersects every other circle at exactly $2$ points. Find in terms of $N$ the minimum and maximum number of disjoint closed regions that can be formed.
Hint: You may wish to check your answers visually for greater $N$, e.g $N=4$.
The answer will be: $$ N(n)=n^2-n+2 $$ This can be proved as: When a new circle is introduced in a collection of $n$ circles, $2n$ new regions will be created (To see this, check the number of parts in which the new circle will be cut by the existing circles). So, $$S(n+1) = S(n) + 2n$$ Solving this recurrence with the initial condition $S(1)=2$ gives the required answer
