Number of regions in space from planes through origin

Question

I've seen that the cake numbers give the largest number of regions that can be created by cutting 3-D space by N planes. I have a variation on that question. How many regions can be created by N planes that all pass through the origin (subspaces)? In 2-D, using lines it looks like 2N regions can be created. For 3D and planes I'm not sure, but I imagine it will be less that the cake numbers.

Answer 1

Let's first plane $p_0$ be $z=0$. Consider the plane $p'_0$ $z=1$. All partition of space with $p_0$, $p_1$,...$p_n$ will create a pattern of lines $a_1$,... $a_n$ on a plane $p'_0$ and vice versa.

For every flat piece on plane $p'_0$ there are two subspaces: one above $p_0$ and one below. So your number is twice the $(n-1)$-th central polygonal number: $$ k = (n-1)^2+(n-1)+2 = n^2-n+2. $$