# Dividing regions of a plane by lines

At most how many regions can be divided by 10 lines on a plane?

This is not homework, this is from a math competition.

I figured out by drawing a picture that with 2 lines I can split the plane into at most 4 regions, with 3 lines I can split the plane into at most 7. I am having trouble generalizing for more lines because there are a lot more possibilities. Any help is appreciated.

• oeis.org/A000124 – Ed Pegg Aug 12 '16 at 23:42
• Hint: For any line n you add you get n additional areas. – Moti Aug 13 '16 at 1:20

## 1 Answer

Based on the comment, the answer is sum of the n sequence + 1: ${n}{(n+1)}/{2}+1$. For 1 you get 2, for 2, 4 and for 3 you get 7, and so on.

• Thank you for your answer! But I'm still confused why that gives the max number of regions. – jw35174 Aug 13 '16 at 15:19
• The reasoning is as follow: every time you cross a line you add a partition. The first partition starts in the starting point in infinity and after that n-1 lines are crossed, to a total of n partitions added by the n's line. – Moti Aug 13 '16 at 15:49
• Thank you again. Is this right? To make the max number of regions possible, make sure no 2 lines are parallel, and make sure thru every point there are $\leq$ 2 lines going thru that point. – jw35174 Aug 13 '16 at 18:16
• This seems to be correct. – Moti Aug 13 '16 at 19:33