From

Pizza Cut, we know the formula for maximum partitioning (pieces) of the circle, given

*n* straight lines (cuts).

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- Determine the maximum number of regions of the plane produced by
*n* intersecting circles.

- Determine the maximum number of regions of the plane produced by
*n* intersecting ellipses.

- Determine the maximum number of regions of
**space** produced by *n* intersecting spheres.

First of all, although nikki's formulae may be correct, it falls short of a proof. Here are some ideas I am flirting with:

- For Part 1, it turns out that the answer is the same for n circles and for (n-1) circles and one line. The reason for this is that one may take any configuration with n circles, pick a non-intersection point on one of the circles, and perform geometric inversion about that point. It *may* be possible to perform some kind of induction by reasoning about the addition of a line to an existing configuration.
- The same is true (I think) for Part 2. To see this, take a configuration with n ellipses, and perform an affine transformation so that one of the ellipses maps to a circle. Then perform the inversion as before so that this circle maps to a line. The only question is whether the inversion maps ellipses to ellipses, which I don't have time to think about right now...
- The same applies to Part 3, but with (n-1) spheres and one plane.