POW #4 Cutting the pie
During a pie cutting experiment, you are instructed to make various cuts into a single circle. Making one cut along the middle will result in two separate pieces and making a second cut will result in four separate pieces. When continuing to cut the pie, we are looking to make the maximum amount of slices(pieces), no matter what size each piece is. This means that each of the cuts does not have to go directly through the center and there may be multiple intersection points on the pie. When solving this problem, we are presented with the task of seeing what is the maximum amount of slices you can create with the minimum amount of cuts.
When solving the problem, I first started off by making small circles and experimenting with the different amounts of cuts that I though to be the maximum. My experiments varied in the amount of cuts I would make, the amount of intersections I made, and the outcome also varied (the amount of slices). After doing various experiments it seemed the more intersections that you made, the more slices that you were able to create. I decided to create a table with different sections. One of which measured the amount of cuts I made, then the amount of slices it created. In evaluation of those two, the the third section looked at the number of intersections that resulted from my cuts. Once I started getting into pies that needed a lot of cuts, I decided to continue my discoveries on the board. This way, I was able to make my pictures really big. This is important because when creating the maximum amount of pieces, some of them are very small. This allows me to have more room to work with in these small sections.
When finding the solution to the problem, I figured out that when using the maximum amount of cuts, the change between each one increased by one each time you increased the number of cuts. For example, with two cuts you would end up with four pieces. With three cuts you would end up with seven pieces for a change of three. With four cuts, you would end up with eleven pieces for a change of three. When looking at the maximum of cuts I was able to conclude that where x represents the amount of cuts, y represents the amount of pieces, and z stands for the number of intersections, you can solve for z by plugging in variables for the equation x+z+1=y. You can see the answers I got for each amount of cuts on the table underneath. The dotted lines are where I decided to stop the tests, and follow the pattern to write out the answers up to 10. My theory for why this pattern is happening is because with every cut you make it gives one more chance of another intersection. This is why the amount of intersections rises along with the # of pieces, because the number of pieces rely on the number of intersections.
I believe I deserve a 10 on this problem because I worked hard to make sure I had diagrams of the max amount of pieces as well as experimenting with different theories. I also was able to come up with an answer and gave a well worded write up to describe my progress with the problem. I feel as if I used the Habit of a Mathematician "solve a simpler problem", because I took each circle one by one. I didn't worry about making a pattern right out of the gates. Once I had all the information together it made it very easy to solve the problem. This process also required me to "look for patterns". I had to compare the results in order to find the pattern that I was able to come across.