[Geometry]
Problem: A cube has edges of length $n$, where $n$ is an integer. Three faces, meeting at a corner, are painted red. The cube is then cut into $n^{3}$ smaller cubes of unit length. If exactly 125 of these cubes have no faces painted red, determine the value of $n$.

Solution: If we remove the cubes which have red paint, we are left with a smaller cube with measurements, $(n-1) \times(n-1) \times(n-1)$

Thus, $(n-1)^{3}=125$

$$
n=6 \text {. }
$$

Final answer: 6