I came across the following question:
In front of you is a large pizza. The pizza may only be cut in straight lines. What is the maximum amount of pieces that the pizza can be cut into using $10$ straight cuts?
They are expecting the answer $56$. This follows from the series where the maximum number of parts is equal to the number of parts with one slice less, plus the number of slices you are allowed to make.
number of cuts 0 1 2 3 4 5 6 7 8 9 10
number of parts 1 2 4 7 11 16 22 29 37 46 56
This can be turned into $c\times(c+1)/2+1$ where $c$ is the number of cuts.
However, because the question is about pizzas and not about circles, you would be able to make pizza parts in $3$ dimensions. When trying to find a series by hand I got stuck at three cuts (which I believe can yield $8$ pieces).
Is there a series or a closed formula that describes the maximum number of parts when slicing a pizza in $3$ dimensions?