jmc

Assume we have a scalene triangle $ABC$. Arbitrarily, let $12$ be the height to base $AB$ and $4$ be the height to base $AC$. Due to area equivalences, the base $AC$ must be three times the length of $AB$. Let the base $AB$ be $x$, thus making $AC=3x$. Thus, setting the final height to base $BC$ to $h$, we note that (by area equivalence) $\frac{BC\cdot h}{2}=\frac{3x\cdot 4}{2}=6x$. Thus, $h=\frac{12x}{BC}$. We note that to maximize $h$ we must minimize $BC$. Using the triangle inequality, $BC+AB>AC$, thus $BC+x>3x$ or $BC>2x$. The minimum value of $BC$ is $2x$, which would output $h=6$. However, because $BC$ must be larger than $2x$, the minimum integer height must be $\boxed{5}$.