Mathematics........Help!!!?

Question

Can anyone tell me an answer 2 this question.It goes like this:
When whe draw a chord on a circle with 2 points 2 regions are formed. Next we add a point and join this point to the other 2 points in a straight line. Then there are 4 regions formed. Next there are 4 points then 8 regions are formed. When 5 points are there 16 regions are formed. When 6 points are added we presume that 32 regions will be formed, but its 31. My question is how to find the number of regions formed when 'n' number of points are given??

Answer 1

Neat question!

Let n be the number of points on your circle.
When you add point (n+1), here's what happens:
adjacent
-if n=1, you form one chord to a point 1-away-from-adjacent
- if n>1, you form two chords to points 1-away-from-adjacent
- if n=3, you form one chord to a point that's 2-away-from adjacent
- if n>3, you form two chords to points 2-away-from-adjacent
- if n=5, you form one chord to a point 3-away-from-adjacent
- if n>5, you form two chords to points 3-away-from-adjacent
... and so on and so forth.

New chords to adjacent points will only cut through one pre-existing region, so they add 1 region to the total.

New chords to 1-away-from-adjacent points add (n-1) regions to your total.

New chords to 2-away-from-adjacent points... I think they add (2n - 5) regions to your total, but I could be wrong. I'll let you do the verifying.

I haven't bothered figuring out the formula for chords to 3-away points (or those further away than that). I'm sure there's a formula, and there should be a pattern for the formulas for n-away chords, but it's hairier than I want to figure out at this time.

Note that there can't be any set formula for the answer you want, because depending on where you add a new point, you could have a new chord pass through the intersection of two old chords. For each time that happens, you'll get one fewer regions than your "formula" answer would indicate.

Anyway, this isn't a full answer, but hopefully it'll help you get to where you need to go.

Answer 2

I thought u'er going ask about Bertrand paradox.

We need a diagram here.....