# Maximum number of pieces of pizza when making 7 cuts

If we have a circular pizza then the maximum number of pieces we can get by making $7$ cuts in it?

The fact that I know the solution only got me the way to find it but it was like a kid trying to make all those cuts on a piece of a paper. is there some logical way of getting the answer?

• Same as for the plane, math.stackexchange.com/q/209672 – user147263 Feb 15 '16 at 18:48
• Hope this helps: I once went into a pizza shop and ordered a pie to go. When it finally came out, the counterman asked me, "Do you want it cut into six or eight pieces?" I thought about it a bit, then answered, "Better cut it into six pieces: I don't think I could eat eight." – Senex Ægypti Parvi Feb 15 '16 at 23:28

One line splits the plane in two parts. If we add a second line, we split it in four parts. With a third line that crosses both the previous line we split the plane in seven parts. Till now, the situation is the following: $$\begin{array}{ccccc}\text{number of lines:} & 0 & 1 & 2 & 3 \\ \text{number of parts:} & 1 & 2 & 4 & 7\end{array}$$ Any further line may cross the previous $n$ lines, giving $(n+1)$ extra parts and no more.
It follows that the number of parts is a second-degree polynomial in the number of lines: since $\sum_{k=1}^{n}k = \frac{n(n+1)}{2}$, we have that $n$ lines split the plane in at most $$\frac{n(n+1)}{2}+1$$ parts, so seven lines may split the plane (or the circle) in $\color{red}{29}$ parts.