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.

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

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.

  • $\begingroup$ Thank you for your answer! But I'm still confused why that gives the max number of regions. $\endgroup$ – jw35174 Aug 13 '16 at 15:19
  • $\begingroup$ 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. $\endgroup$ – Moti Aug 13 '16 at 15:49
  • $\begingroup$ 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. $\endgroup$ – jw35174 Aug 13 '16 at 18:16
  • $\begingroup$ This seems to be correct. $\endgroup$ – Moti Aug 13 '16 at 19:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.