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Given $n$ circles of possibly different radii, how many distinct regions can there be? For small $n$, I can work it out with pictures. (I'm pretty sure $n=4$ can yield 13 distinct regions, but not positive.)

Just curious what an approach to the problem could be.

E.g. I have considered that is could be a quadratic sequence based on skimpy evidence, but don't know why that would be, or how to prove it.

I would also be interested to know if there was a way of doing this problem with regular $n$-gons. (I am assuming that there would be more regions, but am not sure.)

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    For more than 2, the proof for circles goes through. Note I left out the exterior above, so two n-gons can give $2n+2$. Two n-gons can intersect in $2n$ points, so if you have $m$ already, the $m+1^{\text{st}}$ can intersect in $2mn$ points, adding $2mn$ regions. In total, for $m$ n-gons, we get $2+m(m-1)n$ regions, where for circles $n=1$ instead of $\infty$.2011-05-18

2 Answers 2

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OEIS gives the number of regions for circles. It gets 14 for 4 circles, including the exterior. The general formula is $n^2-n+2$. Allowing different size circles doesn't help. The proof is by induction. With one circle there are $2$ regions. Assuming the formula is true for $n$, the new circle can cross old ones in at most $2n$ points. Each segment can cut an existing region in two parts, adding $2n$ regions. So the maximum number of regions for $n+1$ circles is $n^2-n+2+2n=(n+1)^2-(n+1)+2$

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    Not a problem! I'm sorry i$f$ I came across as scolding...I certainly didn't intend to...I just "look out $f$or others" - in terms o$f$ acceptance of answers (I'd do the same for my own posted answers, when it's clear it helped, but that would be "tooting my own horn"...2011-05-18
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The source quoted by Ross Millikan (OEIS) gives different formula:

$n^2+n+2$

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    @Ross, @necod: please see my comment above this answer for more details.2011-05-18