It’s an exercise in induction to prove that lines in general position divide the plane into latex regions, and this number of regions is the maximal possible. Here is a partition that realizes :
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 to be the smallest integer such that for any integer there is a partition of the plane into regions by lines. So, , and of course and . Here are a few more values of , with examples in lieu of proofs.
The last one, unattainability of 23 with 8 lines, isn’t easy to prove.
Does 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 for which there is a partition of (projective) plane by lines into 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 out of Martinov’s formulas.