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July 27, 2016
Posted by **bob** on Friday, January 7, 2011 at 1:00am.

ex. 2pts has 2 regions,

3pts has 4 regions,

5pts has 17 regions

- Math -
**MathMate**, Friday, January 7, 2011 at 8:57amFor 4 points, there are 8 regions.

For 5 points, there are 16 regions (not 17).

The numbers form the sloan's sequence A027423 (1,2,4,8,16,30,60,96,...)

Eric Weinstein has studied this problem and provided a formula in Mathworld. It seems a little complex for odd values of n. The formula for n even is a simple polynomial.

Note that the formula given in the link below refers to the subdivision of a*polygon*. For circular division requested, you need to add k, the number of points on the circumference.

Also, the formula starts with n=1 for a triangle, so you need to substitute n=k-2 in the formulas.

For example, for k=4 points, substitute n=k-2=2 in the formula to get 4, but*add*k to get 8 as your answer for a circular subdivision.

http://mathworld.wolfram.com/RegularPolygonDivisionbyDiagonals.html