Maximum possible parts of a cake if cut n times

Question

In how many maximum/minium parts can a round cake($\large\text{ cylinder}$) be divided with $n$ cuts when each cut is necessarily a straight line?

What would be the case if we have a circle,sphere,toroid,etc

Answer 1

This is known as the lazy caterer's sequence, the number is $\frac{n(n+1)+2}{2}$. The idea is to use Euler's Formula $E-V+F=2$ and the fact that the number of edges increases $1$ more than the number of vertices in each line.

Answer 2

I'd like to say $2^n$ in dimension $n$ for $n$ cut, because you can always find a hyperplane (I consider a cut being a hyperplane) cutting all the ones already drawn... At least, with one cut (line cut, of course) you can only double the number of parts, so $2^n$ is an easy upper bound, and the previous argument should work properly written.

For the plane case (usual cake without horizontal cut), I'd like to say $2n$, but you can do much more, cutting a bit away from the center : with 3 cuts you get not 6 but 7 parts.

With one cut you can at best double the number of parts, but of course you should try to keep the maximum number of part intersecting one half of the cake (cutting implies not cutting in one half plane !)...

Well, merely some thoughts.