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Slicing Up a Circle

Date: 03/22/2001 at 04:47:14
From: Don Adams
Subject: Slicing up a circle

I'm doing a unit on Mathematical Induction, and I remember seeing a 
problem about slicing up a circle. The guys in the department 
remember seeing the problem, but no one remembers exactly how it 
goes. We believe it is something like this:  

Come up with a formula that will give the maximum number of pieces 
with n number of straight slices of the circle, i.e. 

1 slice, 2 pieces;
2 slices, 4 pieces;
3 slices, 7 pieces;
4 slices, 11 pieces.

I seem to remember that this suggests a formula that appears to work, 
but as n gets larger it doesn't.  Can you help me?

Date: 03/22/2001 at 14:25:36
From: Doctor Rob
Subject: Re: Slicing up a circle

Thanks for writing to Ask Dr. Math, Don.

I used to use the following example when I taught induction, as a 
warning not to jump to hasty conclusions.

Take n points on the circumference of a circle, and connect them up
in all possible ways with straight lines. Now count the maximum number 
N of regions so formed.

   n    N
   1    1
   2    2
   3    4
   4    8
   5   16
   6   ??

It appears at first glance that N = 2^(n-1).  That would give N = 32
for n = 6, but the correct answer is 31. For n = 7, the correct number 
is 57, not 64, and so on. It turns out that N is actually a fourth 
degree polynomial in n.

For a more thorough explanation, see

  Counting Regions Formed by Chords of a Circle   

- Doctor Rob, The Math Forum   
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Polynomials

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