Question: How many pieces can a solid torus be cut into with three (affine) planar cuts?
A google search will quickly reveal that the answer is thirteen, as can be read about here. The picture below displays this solution. However, despite being able to look at the diagram and confirm that thirteen pieces is indeed possible, I do not see how one would prove this is the maximum number of cuts. Furthermore, I have found the following general formula for $n$ cuts: $f(n)=\frac{1}{6}(n^3+3n^2+8n)$, but have not been able to find a proof for this either.
Better Question: How does one prove that thirteen is the maximum number of pieces with three cuts, and that $f$ provides the number of pieces with $n$ cuts?