jmc

The possible medians after $n$ is added are $6$, $n$, or $9$. Now we use casework. Case 1: The median is $6$ In this case, $n<6$ and $$\frac{3+n+6+9+10}{5}=6\Rightarrow n=2$$ so this case contributes $1$. Case 2: The median is $n$ We have $6<n<9$ and $$\frac{3+6+n+9+10}{5}=n\Rightarrow n=7$$ so this case also contributes $1$. Case 3: The median is $9$ We have $9<n$ and $$\frac{3+6+9+n+10}{5}=9\Rightarrow 17$$ so this case adds $1$. In all there are $3\to \boxed{3}$ possible values of $n$.