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### Slicing Up a Circle

```
Date: 03/22/2001 at 04:47:14
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

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
http://mathforum.org/dr.math/problems/shah5.19.98.html

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Polynomials

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