Consider the following diagram and fill in the table: It's a diagram of circles with points around the circumference based off of the table. The table reads
# of Points: 2, 3, 4, 5, 6
# of Regions: 2, 4, 8, ?, ?
For the 5 points I got 16 regions
when i use inductive reasoning there should be 32 regions for 6 points but if you draw it out there are only 30 regions. Why is this so?
Even though you posted this question 4 times now, you are probably not getting any replies because, speaking for myself, I don't know what you mean by "region".
Secondly, are we dealing with one circle? You mentioned circles, so how many.
Or, ... is there a single circle with, let's say, 4 points on it, another with 5 etc?
Looking at the # of regions the answers appear to be powers of 2
So I would guess that
Number(n) = 2^(n-1) , where n is the number of points on the circle.
so 5 points ---> 16 regions
6 points ---> 32 regions
7 points ---> 64 regions, etc
This seems to be a question dealing with number of subsets.
e.g. given points A,B,C,D
I can form 2^4 or 16 subsets
{} -- 1
A,B,C,D -- 4
AB,AC,AD,BC,BD,CD --6
ABC,ABD,ACD, BCD -- 4
ABCD -- 1 for a total of 16
Can you related this to the number of regions?
To understand why there are only 30 regions when 6 points are used, let's analyze the problem based on the given information. The table shows the relationship between the number of points on the circumference of the circles and the number of regions created by connecting those points.
We are given that when there are 2 points, there are 2 regions. This suggests that the first circle forms a single region, and the addition of the second point divides that region into two.
When we have 3 points, we have 4 regions. To determine this, we can imagine adding the third point to the two points already present. Connecting the third point with the other two forms three new regions. Additionally, the three points themselves create another region, giving us a total of 4 regions.
With 4 points, we have 8 regions. Here, we can again use the principle of connecting the new point to the existing ones. The new point will intersect with the previous lines and create three new regions. Connecting the new point with each of the other three points forms three more regions. Finally, the four points themselves create two more regions. This gives us a total of 8 regions.
Now, let's consider 5 points. You mentioned that by drawing it out, you found 16 regions. This means we have 8 new regions created by adding the new point, which is consistent with the pattern we observed previously. Connecting the new point with each of the five previous points will create five more regions, resulting in a total of 13 regions. In addition to these, the five points themselves create three regions (one big region with two smaller ones inside), giving us a total of 16 regions.
At this point, we can conclude that the pattern is not consistent. If we assume that each additional point adds the same number of new regions, we would expect 32 regions for 6 points. However, as you observed, the actual number of regions is only 30.
The reason for this discrepancy lies in the way the new point intersects with the existing lines. When the sixth point is added, it intersects with two existing lines, dividing each of them into two separate arcs. This creates two regions, but these regions overlap with each other, decreasing the total number of regions from the expected 32 to 30. This overlap occurs because the outermost arcs formed by connecting the sixth point with the first and fifth points cross each other.
In conclusion, the discrepancy between the expected 32 regions for 6 points and the observed 30 regions arises due to the overlapping regions caused by intersecting lines.