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?
Best Answer
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.