"What is the maximal number of regions that can be obtained by joining n points around a circle by straight lines? The first few values of this sequence are 1,2,4,8,16,... Here's a hint: the next value is not 32."

Hope that nobody will take offence if I use this space as a journal of my thoughts on this problem:

- The next value in the sequence is 31, intriguingly.

- If there are https://nrich.maths.org/tex2gif/tex2gif.php?tex=n distinct points around the circle, the number of chords is https://nrich.maths.org/tex2gif/tex2...tex=%5En%20C_2.

- Having seen that enumerating intersections was useful in a previous simpler but similar problem (see https://nrich.maths.org/discus/messa...tml?1248745070), I have been trying to enumerate the number of intersections that do *not* occur on the circumference of the circle. (Clearly, there are https://nrich.maths.org/tex2gif/tex2gif.php?tex=n chords between adjacent points, which create https://nrich.maths.org/tex2gif/tex2...ex=n%20%2B%201 regions: https://nrich.maths.org/tex2gif/tex2gif.php?tex=n regions closest to the circumference, and one central region. I hope that if I can enumerate the number of intersections that occur within the circle, I can deduce how this affects the division of this central region.) If we label a point on the circle https://nrich.maths.org/tex2gif/tex2gif.php?tex=A, and work our way around the circle in one direction only, we may label the subsequent points https://nrich.maths.org/tex2gif/tex2...0C,%20%5Cldots. I will use the https://nrich.maths.org/tex2gif/tex2gif.php?tex=%3C and https://nrich.maths.org/tex2gif/tex2gif.php?tex=%3E symbols to denote position with respect to this order, with https://nrich.maths.org/tex2gif/tex2gif.php?tex=A defined to be the least member.

The chord linking the points https://nrich.maths.org/tex2gif/tex2gif.php?tex=X and https://nrich.maths.org/tex2gif/tex2gif.php?tex=Y I will denote https://nrich.maths.org/tex2gif/tex2gif.php?tex=X%20Y. Suppose that the chord https://nrich.maths.org/tex2gif/tex2gif.php?tex=YM intersects with the chord https://nrich.maths.org/tex2gif/tex2gif.php?tex=XZ where https://nrich.maths.org/tex2gif/tex2...ex=X%20%3C%20Z within the circle; then https://nrich.maths.org/tex2gif/tex2gif.php?tex=YM must adhere to the following:

https://nrich.maths.org/tex2gif/tex2...%20Y%20%3C%20Z and (https://nrich.maths.org/tex2gif/tex2...ex=M%20%3C%20X or https://nrich.maths.org/tex2gif/tex2...ex=M%20%3E%20Z).

But how to enumerate these internally intersecting chords? I suppose that the points https://nrich.maths.org/tex2gif/tex2gif.php?tex=X and https://nrich.maths.org/tex2gif/tex2gif.php?tex=Z in the above intersecting chord must not be adjacent. There are https://nrich.maths.org/tex2gif/tex2...28n%20-%201%29 of qualifying chords that may play the role of https://nrich.maths.org/tex2gif/tex2gif.php?tex=XZ. I suppose that I could enumerate the number of such pairs of chords (and therefore the number of internal intersections), noting that the intersecting chord pair will be counted twice in the calculation, since it may play the role as either https://nrich.maths.org/tex2gif/tex2gif.php?tex=XZ or the role of https://nrich.maths.org/tex2gif/tex2gif.php?tex=YM.

I'm not sure if any of the last paragraph will make much sense to anyone else, but I hope that it will to me in the morning, when I shall think about this some more...

Any thoughts, or hints much appreciated. (Talking)