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

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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.

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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 $A=21$.

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  • $\begingroup$ 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 :( $\endgroup$ – UnidentifiedX Feb 24 at 11:34
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I get 21:

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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"
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  • $\begingroup$ 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? $\endgroup$ – UnidentifiedX Feb 22 at 2:44

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