Number of lines required to split a plane into N regions

Question

I was wondering if there was any theorem out there that talks about the number of lines required to split a plane into any given number of regions. I don't really have much of a mathematical back ground but I'm beginning to learn some, so I hope that the dumbness is excused.

Answer 1

Take a look at this.

The upshot is that $n$ lines can cut a plane into at most $\frac{1}{2}(n^2+n+2)$ regions.

So, to cut the plane into $N$ regions, you will need at least $$ \frac{-1+\sqrt{8N-7}}{2} $$ lines to do it (when this is not an integer, you round up to the nearest integer to find the minimal number of lines you need).

By the way, http://oeis.org/A000124 has a lot more links and references to investigate.