Slicing Up a CircleDate: 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 http://mathforum.org/dr.math/problems/shah5.19.98.html - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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