\%EA360 * Find the number of positive integer solutions $(m, n, r)$ for the indeterminate equation
$$
m n+n r+m r=2(m+n+r)
$$
If $m, n, r$ are all greater than 1, then $m n + n r + m \geqslant 2 m + 2 n + 2 r$, with equality holding only when $m = n = r = 2$.

If $m = 1$, then $n = 2 + n + r$, which simplifies to $(n-1)(r-1) = 3$. Therefore, $n = 2, r = 4$ or $n = 4, r = 2$.
The cases where $n = 1$ or $r = 1$ are similar.
Thus, (1) has a total of 7 solutions, namely $(2,2,2), (1,2,4), (1,4,2), (2,1,4), (1,1,2)$, $(2,4,1), (4,2,1)$.