January 25, 1999
Problem of the Month

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 to the Problem:

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

Send any comments or questions to: David Pleacher