# Different geometry problem

Acolyte
Posers and Puzzles 30 Oct '02 23:18
1. Acolyte
30 Oct '02 23:18
When I posted the first one I meant to post a slightly different problem. Here it is:

Draw a circle. Draw n points on the circumference. Join up the points with straight lines.
How many regions can you divide the circle into?
2. Acolyte
02 Nov '02 16:18
3. 03 Nov '02 11:09
this would be the same problem as the other one, except that the
number of lines has to be calculated first: the combination of n
elements in groups of two: n x (n-1) / 2

And we substitute that into the formula of the other problem

number of regions = 1 + [N x (N+1)] / 2 (the other problem)
with N= n x (n-1) / 2
gives : 1 + [ n x (n-1) x ( n² - n + 2)] / 8

lets try with
2 points: 1 line, 2 regions
3 points: 3 lines 7 regions
4 points: 6 lines, 22 regions
etc.....

or am I talking nonsens (again)?

sin.
4. Acolyte
03 Nov '02 16:06
Your first answer was correct, but this answer isn't, I'm afraid. You can't make 7 regions with
just 3 points. Hint: the lines aren't arbitrary, as they were in the first problem.
5. 03 Nov '02 16:19
you are right (nothing to be afraid of, LOL). I was a bit naive. There
are two important differences with the first problem:

1) In the first problem I could imagine the circle outside ALL
intersections. So the area's between intersections laying outside the
circle have to be deducted.

2) in the first problem, all lines could have intersected at different
points; Here there are a lot of coinciding intersections.

Too complicated to answer just like that (at least starting from this
angle). Back to the study .... Anyone else?
6. Acolyte
03 Nov '02 17:15
I find the easiest way to think about this is by induction. Put a few points down, and see
what hapens when you add one more point.
7. richjohnson
TANSTAAFL
04 Nov '02 21:02
2 points = 2 regions.
3 points = 4 regions.
4 points = 8 regions.
5 points = 16 regions.
6 points = 30 regions?? Am I counting wrong?
8. Acolyte
04 Nov '02 23:54
For 6 I can make more than 30. Don't put the points in a regular hexagon.
9. richjohnson
TANSTAAFL
05 Nov '02 00:09
with irregular placement I count 31 regions - still not the expected 32.
10. Acolyte