I'm reading Concrete Mathematics: A Foundation for Computer Science (for my own amusement), and after working out the recurrence for the number of three-dimensional regions that can be defined by $n$ different planes, I got interested in another question, more geometrical than combinatoric in nature, which I have no idea how to begin solving.
The recurrence mentioned:
Given some regular three-dimensional figure such as a cube, the number of pieces $P_n$ it may be cut into by $n$ slices (planes) satisfies the following recurrence (spoilered because it is a great problem to work out yourself):
$$P_n=P_{n-1}+L_{n-1}$$ $$L_n=L_{n-1}+n$$ $$P_0=1$$ $$L_0=1$$
($L_n$ as defined above is the maximal number of two dimensional regions that may be formed on a plane by $n$ lines.)
$P_4=15$, so for instance it is possible to cut a large cube of cheese into 15 pieces using 4 slices.
My question:
Is it possible to cut a cube into $15$ pieces of equal volume using $4$ straight cuts?
How can I go about solving this? Other than getting a large lump of clay and a knife to experiment with and observe, is there any geometric method that would allow me to work out the necessary cuts and prove they are correct?
A follow up question of interest: it is obviously not possible to have all 15 pieces the same shape (though with $3$ cuts it is trivially easy to make $P_3=8$ identical pieces). However, is it possible to have 2 or more of the 15 pieces identical? And inversely, is it possible to have no pieces identical (but all have the same volume)?
If it is possible, is this possible for all $n$? In other words, for any $n \ge 0$, is it possible to cut a cube into $P_n$ pieces of equal volume?