I proposed this problem in a birthday team competition I wrote, but at the time I asked for its equivalent in 18 dimension.

Today I’ll explain it in 3 dimensions, and I want to point out that 3 is the perfect number for today’s occurrence.

Now, how many parts of space (finite or infinite) can n planes generate at most?

To answer such question, we’ll use the extremal principle.

First of all, to have the maximum parts of space every two planes must intersect in a line, every three planes must intersect in exactly a point and no four planes can intersect in a point.

Now, let’s fix a reference system, and, for each part of space, we consider its lowest vertex. There is a bijection between the parts of space and such vertexes, so we say that there are parts.

But we are not considering the ones that go infinitely downwards.

We can project them onto a plane and consider the problem in 2 dimensions: since there are n planes, they will generate n lines on the plane we’re projecting to.

By a similar reasoning, there are parts which correspond to a lowest point, and we project the others onto a line. Since there are n lines, each forms one point on the line, meaning there are finally n+1 parts to consider.

Hence, the toal number of parts is

It’s easy to generalize for k dimensions and see it’s the sum of the binomial coefficients for all