# Partition of the plane by lines

It’s an exercise in induction to prove that $n$ lines in general position divide the plane into latex $M_n=n(n+1)/2+1$ regions, and this number of regions is the maximal possible. Here is a partition that realizes $M_3$:

Three lines can also divide the plane into 6 regions instead of 7: this happens if the triangle collapses to a point, or if two of the lines are made parallel. However, 3 lines can never divide the plane into 5 regions.

Define $\mu_n$ to be the smallest integer such that for any integer $m\in [\mu_n, M_n]$ there is a partition of the plane into $m$ regions by $n$ lines. So, $\mu_3=6$, and of course $\mu_2=3$ and $\mu_1=M_1=2$. Here are a few more values of $\mu_n$, with examples in lieu of proofs.

The last one, unattainability of 23 with 8 lines, isn’t easy to prove.

Does $\mu_n$ have a closed form? 2,3,6,8,12,15,18,24 is not in OEIS, unlike the upper bound.

Update: the 1993 paper Classification of arrangements by the number of their cells by Nicola Martinov gives a complete description of the pairs $(n,f)$ for which there is a partition of (projective) plane by $n$ lines into $f$ regions. In the affine version considered above we should simply says that the ideal line is also a part of arrangement; thus, 3-line affine arrangements correspond to 4-line projective arrangements, etc. However, it is not entirely trivial to get $\mu_n$ out of Martinov’s formulas.