# Finding the greatest number of regions on a plane

There are 3 straight lines and two circles on a plane. They divide the plane into regions. Find the greatest possible number of regions.

This is the best that I could come up with an I don't really know how to get even more regions. Using some mental calculations, I figured that I could edit the picture and make it into 20 regions. However, the answer key gave me 21 as the answer. Is there a way to find out the number of regions on a plane without using a calculator? Is there a formula for it? Please help.

The number $$21$$ for a general configuration with maximal number of intersections follows from Euler’s formula. There are $$18$$ vertices ($$3$$ from line/line, $$12$$ from line/circle, $$2$$ from circle/circle intersections, and $$1$$ at infinity). There are $$37$$ edges ($$21$$ from lines, $$16$$ from circles). So $$18 - 37 + A=2$$ and $$A=21$$.

• can you please elaborate more for me? How do you find the different vertices? I’m really sorry but I’m just 12 and I’m trying to prepare for my math competition :( – UnidentifiedX Feb 24 at 11:34

I get 21:

There can be only six regions "at infinity" (as defined by the straight lines).

• 6 "at infinity"
• 2 "local, but outside circles"
• 8 "entirely within red circle"
• 5 "within green circle but not inside red circle"
• Can you explain how you get 21? I understand the part of 'infinity' but where did you get the 21 from? Maybe you can help me number the number of regions? – UnidentifiedX Feb 22 at 2:44