In the book "Build Your Brain Power" by Wootton and Horne, they mention the lazy caterer's problem, asking for a way to cut a circular cake into 8 equally sized pieces with 3 cuts. Clearly since the maximum number of possible segments is 7 for the $n=3$ case, that is impossible.
But, I was nonetheless wondering about this: is it possible, for any $n\geq 3$ (3 cuts or more) case of the lazy caterer's problem, for each of the resulting areas to be equal? How would one prove or disprove this?
A cake is a 3D object, and it is perfectly possible to cut it into 8 equally-sized pieces, but size should be understood as volume, not 2D area as seen from above. Each of the three cuts should be across a different dimension. Note that the description of the cake as circular is misleading, it is actually cylindrical, but describing it as such in the problem would give away the answer.