I noticed that $3$ lines cannot divide the plane into $5$ regions (they can divide it into $2,3,4,6$ and $7$ regions). I have a line of reasoning for it, but it seems rather ad-hoc. I also noticed that there are no "gaps" in the number of divisions that can be made with $n=4$, i.e we can divide a plane into $\{2,3,\cdots,11\}$ using $4$ lines.
Is there anyway to know that if there are $n$ lines with $x$ being the possible number of divisions $\{ 2,\ldots, x,\ldots x_{max} \}$ where $x_{max}$ is the maximum number of divisions of the plane with $n$ lines, then what are the $x \lt x_{max}$ which are not possible?
For e.g, $n=3$, $x_{max}=7$, $x=5$ is not possible.
Best Answer
See if you can find the paper, I N Schnurnikov, Into how many regions do $n$ lines divide the plane if at most $n-k$ of them are recurrent? The abstract says, inter alia,
Thus, a new proof of N. Martinov’s theorem is obtained. This theorem determines all pairs of integers $(n, f)$ such that there is an arrangement of $n$ lines dividing the projective plane into $f$ regions.
Or, if you read Bulgarian, look for Nikola Martinov, Solution of a Federov-Grünbaum problem, Annuaire Univ. Sofia Fac. Math. Inform. 87 (1993), no. 1-2, 73–85 (1999), MR1745340 (2000k:52020).
Wait a minute - it's also in English: Nicola Martinov, Classification of arrangements by the number of their cells, Discrete Comput. Geom. 9 (1993), no. 1, 39–46, MR1184692 (93g:52008).
EDIT 17 May: See also Oleg A Ivanov, On the number of regions into which $n$ straight lines divide the plane, Amer Math Monthly 117 (Dec. 2010) 881-888. Curiously, this paper does not reference Martinov's work.