AMC10 2005 10A, Problem 11

A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$?

Solution: Since there are $n^2$ little faces on each face of the big wooden  cube, there are $6n^2$ little faces painted red. Since each unit cube has $6$ faces, there are $6n^3$ little faces total. Since one-fourth of the little faces are painted red, $\frac{6n^2}{6n^3}=\frac{1}{4}$ $\frac{1}{n}=\frac{1}{4}$ $n=\fbox{4}$

Answer: 4