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You probably know the following problem:

We have two circular cakes of the same height but unknown and potentially different radii, and we want to cut them into two equal shares. Each cut can only cut one piece of one cake. What is the minimal number of cuts required?

The solution is easily found:

It's $1$ cut. Stack the two cakes and center them, then cut the big one around the small one : Here you go.

While I was thinking about this easy problem, the following question logically arose:

We have $n\in\mathbb{N}^*$ circular cakes of unknown and potentially different radii (hmm yummy) but the same height, and we want to cut them into $p\gt n$ equal shares. Each cut can only go through one piece of cake. What is the minimal number $c$ of cuts required? How do you cut the cakes? The cuts can be done however you want (i.e. lines, curves, etc) and stacking cakes etc is allowed. You are of course not allowed to measure the exact radius. To make it simple, let's say you have a straightedge (unlabelled ruler), an arbitary-precision protractor and a compass.

Any help/thoughts are appreciated.