9.75 A rectangular prism is constructed from small cubes of the same size. Three faces that share a common vertex are painted. As a result, exactly half of the small cubes have at least one face painted. How many small cubes have at least one face painted?

The rectangular prism is constructed from small cubes of the same size. Three faces that share a common vertex are painted. As a result, exactly half of the small cubes have at least one face painted. How many small cubes have at least one face painted?
[Solution] Let the edge length of the small cube be 1, and the size of the rectangular prism be $m \times n \times k(k \leqslant n \leqslant m)$. Clearly, the number of uncolored small cubes is $(m-1)(n-1)(k-1)$. From the given information, we have
$$
m n k=2(m-1)(n-1)(k-1) .
$$

From (1), we can derive $2<k<5$. Thus, the remaining task is to find the integer solutions of the indeterminate equation (1) for $k=3,4$.
When $k=3$, equation (1) becomes
$$
m n-4 n-4 m+4=0 \text {. }
$$

It is not difficult to see that $4<n<8$. Thus, we get three solutions of (2): $(5,16),(6,10),(7,8)$.
When $k=4$, equation (1) becomes
$$
m n-3 m-3 n+3=0 \text {, }
$$

From (3), it is easy to see that $4 \leqslant n<6$. Thus, we get two solutions of (3): $(4,9)$ and $(5,6)$.
From the five solutions obtained from (2) and (3), we know that there are 5 possible values for the number of colored small cubes: $60,72,84,90,120$.