POW 17- Cutting the Pie

Problem Statement-

If you were given a pie what is the maximum number of pieces you can produce from 4, 5, and 10 cuts? Keep in mind, that the slices do not have to be the same size and the cuts do not necessarily have to go through the center of the pie, but the cuts do have to be straight and go all the way across the pie.   Include any diagrams you used to find the solution such as an   In-Out table, or any patterns you found.

Process-

The first thing I did to try to find my solution was to finish the In-Out table given, which already told us the maximum number of pieces that could be made with 1, 2 and 3 cuts. So I drew two circles, and drew in four cuts in one and five cuts in another to find the maximum number of pieces that could be produced. After several circles for each I found that the maximum number of pieces you can produce from four cuts is 11, and the maximum number of pieces for five cuts is 15.
In-Out Table
In-X (Number of cuts) Out-Y (Maximum # of pieces)
1 2
2 4
3 7
4 11
5 16

So instead of trying to do the same thing t find out the maximum number of pieces for 10 cuts, I started looking for a pattern. I found that the difference between four and two is 2, the difference between   seven and four is 3, the difference between eleven and seven is 4, and finally the difference between sixteen and eleven is 5. Based, on these results from my In-Out table I found out that one more value is added to the previous addend to come up with the next value. However, this wouldn't work that well because if I was to find the maximum number of pieces that can be produced from 50 cuts I would have to do a lot of tedious work to finally reach 50 cuts. This is because the independent variable is the number of cuts, not the maximum number of pieces. Finding the pattern was easy, but the challenging part was finding the formula. I tried finding something that had to do with the way the actual pie was cut, like how... [continues]

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