Answer to January 25, 1999 Problem

Circle Problem
from Problem Solving A Handbook for Teachers
by Krulik and Rudnik

What is the maximum number of regions into which 8 chords can divide a circle?


Solution:

The answer is 37 regions.
The maximum number of regions is achieved by using chords which are not not parallel, not concurrent, and not intersecting on the circle.

Look at a table:

Number of
chords		 Number of
regions
1 2
2 4
3 7
4 11
5 16
6 22
... ...
n (n^2 + n + 2) / 2
So, for 8 chords,
(64 + 8 + 2) / 2 = 74 / 2 = 37

Correctly solved by:

1. Kelley Britz Winchester, VA
2. Jeremy Ramsey Winchester, VA
3. Trey Genda Winchester, VA
4. Krista Ramey Winchester, VA