Date: Mon, 12 May 1997 00:04:09 -0400

From: Alexander Bogomolny

Andy:

The question you asked is not at all simple. I'll give you a couple of formulas. See if you need more help. Proceed as follows:

- First of all assume that no three of your chords pass through the same point.
- Note that the problem admits a different wording. x points on a circle form a convex polygon. Disregarding x circular segments between its sides and the circle, the question is into how many regions a convex polygon is divided by its diagonals. If A(x) is the number you are after, and B(x) is the answer to the latter problem, then A(x) = B(x) + x.
- The answer to the problem #2 is obtained by computing the total number of points and the total sum of all angles in thus obtained regions.
- C(x) = x(x-1)(x-2)(x-3)/24 is the number of points of intersection of
all diagonals (this is the hard part!) Let we have r
_{3}triangles, r_{4}quadrilaterals, etc. Then 3*r_{3}+ 4*r_{4}+ ... = 4*C(x) + x(x-2). - Counting the total of all angles, we have
(r
_{3}+ 2*r_{4}+ 3*r_{5}) * Pi = C(x) * 2*Pi + (x-2)*Pi. - Subtracting #5 from #4 gives
2*(r

_{3}+ r_{4}+ ...) = 2*C(x) + (x-1)(x-2)which simplifies to

B(x) = r

_{3}+ r_{4}+ ... = (x-1)(x-2)(x^2 - 3x + 12)/24 - Note that A(6) = 31 and not 32 as you have claimed. Although A(5) = 16 indeed.

Best regards,

Alexander Bogomolny