Brazilian Math Olympiad

The bigger cube has sidelength $\frac{12}{5}$ cm, so the difference between the side-lengths is $\frac{12}{5}-2=\frac{2}{5}$ cm, that is, the red cubes should not have sidelength greater than this length. Cubes with sidelength $\frac{2}{5}$ cm are the natural candidates, so we set a new unit $u=\frac{2}{5}$ cm. Notice that the bigger cube should have sidelength $6u$ and the original cube must have sidelength $5u$. So we need $6^3-5^3=91$ red cubes.