Problem For A Particular Occasion – Mattia Giuri's bizarre blog

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 ${n\choose3}$ 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 ${n\choose2}$ 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

${n\choose3}+{n\choose2}+{n\choose1}+{n\choose0}$

It’s easy to generalize for k dimensions and see it’s the sum of the binomial coefficients ${n\choose i}$ for all $i \leq k$