A bug travels in the coordinate plane moving only along the lines that are parallel to the x-axis or y-axis. Let $A=(-3,2)$ and $B=(3,-2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most 14. How many points with integer coordinates lie on at least one of these paths?

Let the bug move on the integer lattice. The Manhattan distance between $A(-3,2)$ and $B(3,-2)$ is:

We are considering all paths of length at most $14$.

A point $P(x,y)$ lies on at least one path from $A$ to $B$ of length at most 14 if and only if:

where $d$ is the Manhattan distance.

This defines a bounding region in the coordinate plane. By dividing the plane into sectors based on the absolute value boundaries, we count the number of integer lattice points $(x,y)$ satisfying this inequality. The count of such lattice points is exactly $74$.