Example 6. There is a rectangular prism with length, width, and height being positive integers $m, n, r (m \leqslant n \leqslant r)$, respectively. The surface of the prism is painted red and then it is cut into unit cubes. It is known that the number of unit cubes without any red color is equal to the number of unit cubes with only one face painted red (from the league competition).





Note: The translation aims to maintain the original structure and format of the text, including the use of mathematical symbols and the reference to the competition.
The number is $(n-2)(r-2)$.
According to the problem,
$$
(n-2)(r-2)=1985 \text {. }
$$

Obviously, when $n, r<3$, (1) has no solution.
When $n, r \geqslant 3$,
$$
\because \quad 1 \leqslant n-2 \leqslant r-2 \text {, }
$$

and $1985=1 \times 1985=5 \times 397$,

solving it, we get $n=3, r=1987$ or $n=7, r=399$.
$2^{\circ}$ When $m=2$, the number of cubes not touching red is $(r-2)$, and the number of cubes with one red face is $4[(n-2)+(r-2)]$. According to the problem,
$$
\begin{array}{l} 
4[(n-2)+(r-2)]-2(n-2)(r-2) \\
=1985 .
\end{array}
$$
The left side of (2) is even, while the right side is odd, so there is no solution.
$3^{\circ}$ When $m \geqslant 3$, the number of cubes not touching red is $(m-2)(n-2)(r-2)$, the number of cubes with one red face is $2(m-2)(n-2)+2(n-2)(r-2)+2(r-2)(m-2)$, and the number of cubes with two red faces is $4(m-2)+4(n-2)+4(r-2)$.
$$
\begin{array}{l}
\text { Therefore, }(m-2)(n-2)(r-2)+4(m-2) \\
\quad+4(n-2)+4(r-2)-2(m-2) \\
\cdot(n-2)-2(n-2)(r-2)-2(r-2) \\
\cdot(m-2)=1985 . \\
{[(m-2)-2][(n-2)-2][(r-2)-2]} \\
+8=1985 .
\end{array}
$$

That is, $[(m-2)-2][(n-2)-2][(r-2)-2]$

or $(m-4)(n-4)(r-4)=1977$.
$$
\begin{array}{l}
\because \quad m \leqslant n \leqslant r, \\
\therefore \quad m-4 \leqslant n-4 \leqslant r-4 .
\end{array}
$$

L $\quad 1977=1 \times 1 \times 1977=1 \times 3 \times 659$,
$$
\therefore \quad m-4=1, n-4=1, r-4=197
$$

or $m-4=1, n-1=3, r-4=659$
indeed $m-4=-1, n-4=-1, r-4=1077$.
indeed $m=5, n=5, r=10: 1$

indeed $m=5, n=7, r=66$.
or $m=3, n=3, r=1981$.
In summary, the values of $m, n, r$ that meet the conditions are: 1, 3, 1987; 1, 7, 399; 5, 5, 1981; $5,7,663 ; 3,3,1081$.