Circle+Problem

This problem appears in lesson 3 in the 1st edition and Lesson 4 in the 3rd edition:

Draw a circle, put 2 points (dots) on the circle, connect them, and count the number of regions in the circle. Now add a point, connect it to all previous points, and count the number of regions again. Keep adding points and connecting.

Here’s what you get:
 * 1) of points 2 3 4 5 6
 * 2) of regions 2 4 8 ? ?

1. Guess how many regions for 5 and 6 points. 2. Check out 5 and 6. (For 6, it’s better if the points are not perfectly evenly spaced. And you might need a bigger circle.) 3. Are you surprised by the results in either case? (My suggestion:) If so, you might want to check out a circle with 7 points. (I did.)

Sue’s questions: 1. Describe the rule you were using when you made your guess. (The rules I’m thinking of would be something like, if there are this many points, then there will be this many regions.) 2. If the pattern doesn’t fit your rule, there might still be a less obvious rule that would continue to work. How would you go about finding it? (It took me years to figure this pattern out.)