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Question: How many pieces 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, although I can look at the diagram of the cuts 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?

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