Problem+Solution

__The Problem__
By placing 2 dots around a circle and connecting them you get two regions. By placing another dot and connecting all the lines, the circle is divided into four regions. This pattern continues in a geometric fashion until you reach 6 points when SURPRISE! there are 31 not 32 points. Our goal is to find a formula for working out how many regions there would be for n points.

This activity shows how things aren’t always what they seem to be and that patterns may not continue logically. Below are our solutions for finding the real formula. We found 4 different methods of solving, two of which give the formula to work out the number of regions in the circle. Can we assume though that the formula would work on to infinity points? I believe it would, but we need to be wary on because as we saw with our prediction for 6 points the sequence wasn't what it seemed to be. The only way to find out would be through a direct observation with a circle and counting, preferably through some specifically made program.To see how not all things are intuitive as the circle pattern wasn't [|go here].

1st Solution to try: Drawing on paper
Follow this link to see the direct investigation.

This link contains Audrey's effort on finding the number of regions per number of points in a circle through a direct investigation. .

This is the most direct way of observing the sequence. But this solution has its limitations and you can only get up to a few points before the regions become too small to be seen.

2nd Solution: Using Regressions
After using the Method of Finite Difference, Use a graphics calculator to find out the regression of your formula. Regressions are approximates and are not the exact formula.

How to get the quartic regression in a Graphics Calculator Texas Instrument (TI-84)


 * 1) On a TI-84 calculator press >Stat >Edit
 * 2) Enter Data, L1 for points on circle, L2 for Regions that you found through the practical drawing. L1 for X axis, L2 for Y axis
 * 3) Escape (2nd function, quit)
 * 4) Press >Stat >Calc> Quartic Reg
 * 5) Press Enter
 * 6) On the display screen
 * 7) Press L1, L2 >vars, y-vars >Function Y1
 * 8) With these results go >Y= to get the equation

By Jonathan Mackenzie and Daniel Del Pilar

We know we are finding a quartic formula, which should look like this:
 * = x^4 ||= x^3 ||= x^2 ||= x^1 ||= x^0 ||
 * = 1 ||= 1 ||= 1 ||= 1 ||= 1 ||
 * = 16 ||= 8 ||= 4 ||= 2 ||= 1 ||
 * = 81 ||= 27 ||= 9 ||= 3 ||= 1 ||
 * = 256 ||= 64 ||= 16 ||= 4 ||= 1 ||
 * = 625 ||= 125 ||= 25 ||= 5 ||= 1 ||

ax^4+bx^3+cx^2+dx+e

 * x** is the number of points around the circle

How to find the formula using matrices in your calculator using the Graphics Calculator Texas Instrument (TI-84):

1. Press 2nd func > [x^-1] matrix>edit 2. Plug in the given numbers (see diagram above) 3. Press 2nd func > [x^-1] matrix > math > rref( > enter 4. Press 2nd func > [x^-1] matrix > [A] > enter 5. The screen will show 0's and 1's with the formulas on the sides 6. Press [Math] > frac. > enter 7. The formulas will come out as fractions which is easier to input.

The values are the given

e=1
so the formula is:

Example:
Input the numbers in your calculator, where n is the number of points.

For 8 points on the circle a=(1/24)*8^4= 170. 66 b=(-1/4)*8^3= -128 c=(23/24)*8^2= 61.33 d= (-3/4)*8= -6 e=1 SUM= 170.66 + -128 + 61.33 + -6 +1 =99 regions

By Jonathan Mackenzie and Daniel Del Pilar

The 4th Solution
We researched a way to find the number of regions on the internet, in this website [|http://mathforum.org/library/drmath/view/55262.html] we found a single formula, used especially for calculating the number of regions against the number of points in a circle.

r= 1 + C(n,2) + C(n,4)
Where r is the number of regions n is the number of number of points in a circle The combination (C) notation is shown on the diagram below. So for the circle with 6 points You get 31 regions. By Jonathan Mackenzie