A solid 6-by-6-by-6 cube is constructed from unit cubes and the entire outer surface is painted. What fraction of the unit cubes has at least two painted faces? Express your answer as a common fraction.
The corner cubes and edge cubes are the only ones with at least two painted faces. There are $4\cdot12=48$ edge cubes and $8$ corner cubes, for a total of $56$ out of $6^3$. The fraction is  \[
\frac{56}{6^3}=\frac{7\cdot2^3}{3^3\cdot2^3}=\boxed{\frac{7}{27}}.
\]