What is the greatest number of domains(or parts) that n circles could divide the plane?
From many small cases I get the feeling that intersecting circles would provide the greatest number of parts.
Is this recursion right C(n+1) = 2C(n) using the previous statement. Since the new circle intersects all the circles and doubles the parts. Here C(n) is the number of parts for n circles.
How could I prove formally? If I could get an inequality that I will know for sure that I have got the greatest number of parts.
Best Answer
http://oeis.org/A014206 says draw n + 1 circles in the plane; then $a(n)=n^2+n+2$ gives the maximal number of regions into which the plane is divided. There are links and references there.