jmc

If none of the cubes have edge length $2$, then all of the cubes have edge length $1$, meaning they all are the same size, a contradiction. It is clearly impossible to split a cube of edge $3$ into two or more cubes of edge $2$ with extra unit cubes, so there is one $2\times 2\times 2$ cube and $3^3-2^3=19$ unit cubes. The total number of cubes, $N$, is $1+19=20\to \boxed{20}$.