Cutting the Pie

Problem Statement - If you were given a pie (or any other circular-shaped pastry) and were told to cut it a number of ways, what would be the maximum number of pieces you would be able to produce? The cuts you make in the pie do not have to pass the center, the cuts have to just go from one end of the pie to the other. Also, all of the pieces in the pie do not have to be the same size.

Process -

Given the data in the book, I am aware that the most number of pieces you can produce with one cut is two, the most you can make with 2 cuts is 4 and the most you can make with 3 cuts is 7. In hopes of detecting a pattern in the number of pieces, I decide to make circles myself and draw in cuts. I started out with drawing two circles, and making one with four cuts and having one with five cuts.

Shown above are several possibilities for circles with four cuts and five cuts. I took me several circles with four cuts to finally realize that the maximum pieces you can have when you make