December 2024
Problem of the Month
Regions in a Circle
You are given a circle and will select n points on it.
You will then draw all the segments connecting all pairs of these points.
No three segments are allowed to intersect at a common point.
You must then determine the number of regions formed by these segments.
For example, if n = 2, you would draw one segment, and there would be two regions.
If n = 3, you would draw 3 segments, and the number of regions would be four.
In this problem, what is the number of regions formed if n = 6?
(No second chances on this problem)
The solution is:
The answer is 31 regions.
At first, I thought the answer would be 32 because of the pattern:
n = 1: regions = 1
n = 2: regions = 2
n = 3: regions = 4
n = 4: regions = 8
n = 5: regions = 16
But as you can see in the diagram above, the most number of regions that you can get when n = 6 is 31.