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POW: Cutting the pie

The basis of the POW for this week was to answer this theoretical question: if you were to take a circle, and cut it into as many pieces as possible, using "x" amount of divisionary lines, what pattern follows?

To start, we began putting what we knew into graph form, to log results. Below is a table with some basic information:

Number of cuts Max number of pieces
1 2
2 4

This is seen in the initial stages of circle cutting, when things are nice and simplistic. However, things get complicated, when a 3rd cut is introduced.

Number of cuts Max number of pieces
3 7

The question is begged. What causes this pattern to occur? Things get even stranger when we cut four times:

Number of cuts Max number of pieces
4 11

At this, we can finally see a pattern. With every new cut, the maximum number of pieces possible increases in intervals of +1 the difference of the previous answer.

With this, we can conclude that the whole problem can be summarized with the following equation:

M=(N+pM)+1

Or in other words, the maximum number of pieces is equal to the previous number of cuts plus the previous number of max. pieces, plus one. Using this equation, we can find the maximum number of pieces obtained in any number of slices. Here is a table showing off the max. number of pieces obtained from the first 10 examples.

Number of cuts Max number of pieces
1 2
2 4
3 7
4 11
5 16
6 22
7 29
8 37
9 46
10 56