Surface Area and Volume · Superchallenge problem

A box is $2\frac{1}{4}$ inches by $3$ inches by $3\frac{3}{4}$ inches on the inside. It is filled exactly with cubes whose edges are $\dfrac{3}{4}$ inch, and then the lid is closed. How many of the cubes touch the box (its bottom, its lid or one of its sides)?