$\textbf{Cutting Pizza}$
Assume you have a magical pizza in the shape of an infinite plane. You have a magical pizza cutter that can cut an infinite line, but it can only be used $14$ times. To share with as many of your friends as possible, you cut the pizza in a way that maximizes the number of pieces (the pizza is too heavy to be lifted up). How many finite pieces of pizza do you have?
$\textbf{Answer}$
The answer is $78$.
$\textbf{Analysis}$
Let's imagine a circle that contains all the points of intersection inside it. Then, $14$ lines will have $28$ intersection points with this circle. This means that there will be $28$ regions outside this circle. Each of these outside regions is infinite and is joined with one region inside the circle. It follows that the total number of finite regions is the total number of regions created by these $14$ lines minus $28$.
In order to get the maximum number of regions, we should avoid making any two lines parallel and three lines concurrent. In such a case, when a new line is added and intersects with all the existing $k$ lines, this new line will be cut into $(k+1)$ segments. Each segment will divide an existing region into two. This will increase the total number of regions by $(k+1)$. We start with $1$ region when no line exists. Therefore the total number of regions equals $$1+(1+2+3 + \cdots + 14) = 106$$
Hence, the final answer is $106-28=\boxed{78}$.
$\textbf{Note}$
This is a college level math contest problem. However, it does not require any advanced math theorems or formulas. Rather, the core of its solution is a systematic approach to address its key obstacle: how to count the number of infinite slices. (To count the maximum number of all regions is relatively simple and well-known.)