Cutting the pie
Problem Statement
What is the largest number of pieces that can be produced from a given number of cuts in a circle? We had to analyze data from circles we had drawn with 1, 2, 3, 4, 5... cuts to find a pattern between the number of cuts and the maximum amount of pieces produced by the cuts. We lastly had to create an equation explaining how to get the maximum number of pieces with any given amount of cuts made to the circle.
Process
This is where the hard part comes in. My process was really challenging but I tried my best to break and down and start simple. I started by drawing circles and cutting them to get the maximum amount of pieces. I recorded this data and stopped at 5 cuts. Here are the pictures of the 5 circles that I drew.
What is the largest number of pieces that can be produced from a given number of cuts in a circle? We had to analyze data from circles we had drawn with 1, 2, 3, 4, 5... cuts to find a pattern between the number of cuts and the maximum amount of pieces produced by the cuts. We lastly had to create an equation explaining how to get the maximum number of pieces with any given amount of cuts made to the circle.
Process
This is where the hard part comes in. My process was really challenging but I tried my best to break and down and start simple. I started by drawing circles and cutting them to get the maximum amount of pieces. I recorded this data and stopped at 5 cuts. Here are the pictures of the 5 circles that I drew.
After I had all my data my next step was to find patterns in the numbers. I did find one pattern that I will try my best to explain but is a little confusing, in the solution section. After finding the pattern there was only one thing to do... create the equation. I did find one equation but it didn't get me where I needed so I tried again, this time it took me forever to figure it out. As the problem got trickier I researched information on the internet and even asked my dad for help and we found a standard formula to find the sum of any positive sequence of numbers starting from 1. We derived the final formula using parts of the standard formula along with a function principle that my dad and I found in the pattern. After creating the formula using functions and the standard formula we double checked it with the numbers that I already had in my table and it turned out to be a success. The full step-by-step procedure is in the solution section below.
Solution
The solution will be listed in steps.
1. The first step was actually to draw the circles and collect data.
2. I looked at the graph to find patterns. The first pattern I noticed that 2 cuts + the number of previous slices leaves you with the total slices for 2 cuts. The same also applies with 3 cuts, 4 cuts and 5 cuts. (3+4=7) (4+7=11) (5+11=16)
Solution
The solution will be listed in steps.
1. The first step was actually to draw the circles and collect data.
2. I looked at the graph to find patterns. The first pattern I noticed that 2 cuts + the number of previous slices leaves you with the total slices for 2 cuts. The same also applies with 3 cuts, 4 cuts and 5 cuts. (3+4=7) (4+7=11) (5+11=16)
3. Now here is where I did create one equation which was the (S)lices = (I)ntersections + (C)uts + 1
I wasn't pleased with this equation because it was taking me an extremely long period of time to solve for (I)ntersections. So I decided to get rid of the idea of including the number of intersections and create a new equation, but this would involve a little research and help.
4. My dad taught me to create functions to organize the pattern a little better. Here are the functions I got:
- f(1) = 2 = (1+1)
- f(2) = 4 = (2+1+1)
- f(3) = 7 = (3+2+1+1)
- f(4) = 11 = (4+3+2+1+1)
- f(5) = 16 = (5+4+3+2+1+1)
5. After we established the patterns we moved on to see what we could do with the standard formula. The standard formula is 1/2(n(n+1)). In this case we have an additional +1 so our formula would be 1/2(n(n+1))+1 this will simplify to 1/2 (n2+n+2)
6. Our final equation is 1/2 (n2+n+2)
Self-Assessment
Through this POW I feel like I stood on my feet. The reason for this is because I have been trying to find the equation since 9:20am in my math class till now (12:33am). I really did not want to walk into my class empty-handed with no equation. I started out this POW strong and finished strong with major bumps in the middle, not understanding what a function really is, not being able to find the pattern, and most importantly not being able to see the equation. I feel like the best habit of a mathematician I could say I used was start small take apart and put back together. In the beginning I just started off with individual circles and moved slowly through the problem. I eventually came to a point where I had a lot of information but didn't know how to put it together in words for this write-up. I took everything that I knew and rearranged the order of what I did in my head so that it could make more sense to me and this also really helped me understand the problem way better and made me more confident about the equation that I cooked up.
I wasn't pleased with this equation because it was taking me an extremely long period of time to solve for (I)ntersections. So I decided to get rid of the idea of including the number of intersections and create a new equation, but this would involve a little research and help.
4. My dad taught me to create functions to organize the pattern a little better. Here are the functions I got:
- f(1) = 2 = (1+1)
- f(2) = 4 = (2+1+1)
- f(3) = 7 = (3+2+1+1)
- f(4) = 11 = (4+3+2+1+1)
- f(5) = 16 = (5+4+3+2+1+1)
5. After we established the patterns we moved on to see what we could do with the standard formula. The standard formula is 1/2(n(n+1)). In this case we have an additional +1 so our formula would be 1/2(n(n+1))+1 this will simplify to 1/2 (n2+n+2)
6. Our final equation is 1/2 (n2+n+2)
Self-Assessment
Through this POW I feel like I stood on my feet. The reason for this is because I have been trying to find the equation since 9:20am in my math class till now (12:33am). I really did not want to walk into my class empty-handed with no equation. I started out this POW strong and finished strong with major bumps in the middle, not understanding what a function really is, not being able to find the pattern, and most importantly not being able to see the equation. I feel like the best habit of a mathematician I could say I used was start small take apart and put back together. In the beginning I just started off with individual circles and moved slowly through the problem. I eventually came to a point where I had a lot of information but didn't know how to put it together in words for this write-up. I took everything that I knew and rearranged the order of what I did in my head so that it could make more sense to me and this also really helped me understand the problem way better and made me more confident about the equation that I cooked up.