Draw a circle, put a few points on it, connect them all, and count how many regions. If you have no points you get no lines, which gives 1 region - the whole interior of the circle. One point still gets you no lines and one region. Two points gives one lines and splits the interior of the circle into 2 regions. What happens for 3, 4, or 5 points?
It looks like a very familiar pattern, doesn't it?* Now check to see if the pattern holds for 6 points. Hmm...
The goal with this problem is to find an expression (formula) for the number of regions when there are n points. After many years of playing with this problem on and off, I came up with a formula, but I didn't understand why it worked, and so I couldn't be sure if it would always work. I was running an online study group of people working through Harold Jacobs book, Mathematics: A Human Endeavor. This problem appears in the book, and I wanted to be able to lead any discussion that might come up about it. So I needed to understand why my formula worked. I turned to Google, and found lots of answers. They made sense, and I thought I understood.
But my understanding was shallow, and my memory is terrible, so I forgot. That was perfect. I knew I was capable of understanding it, and the next time around, I refused to look it up. With lots of work and my fair share of false starts, I finally figured it out. At first I felt bad about my solution method - I felt like I'd hit the problem with a big hammer, instead of delicately teasing it apart. I'm proud of it now, and I've written a paper describing my solution. I won't post it here, because I think this problem is so worth playing with, I don't want to make finding a solution online any easier than it is. But you're welcome to email me to request it.
I started writing this post over a year ago, and abandoned it, because I couldn't figure out how to say enough to pull people in, without giving too much away. James Tanton has just posted a video that offers a new twist on this problem, which got me thinking about it again, and I'll leave you with that.
* This picture comes from James' video.