Writing It Up · Challenge problem

Look back at the lines problem: $n$ lines cut the plane into $R(n) = 1 + \tfrac{n(n + 1)}{2}$ regions. The same method in space: $n$ planes in general position cut space into $C(n)$ pieces, where the $n$-th plane is itself cut by the others into $R(n - 1)$ regions. How many pieces do $6$ planes make?