Suppose that $*(n) = \left\{ n-2, n+2, 2n, \frac{n}{2} \right\}$. For example, $*(6) = \{4, 8, 12, 3\}$. For how many distinct integers $n$ does $*(n)$ have exactly three distinct elements?
To have three distinct elements, two of $n-2$, $n+2$, $2n$, and $\frac{n}{2}$ must be equal. It is clear that $n-2$ can never equal $n+2$. However, any other equality pairing among the four is possible, so we simply check for distinctness. If $2n = \frac{n}{2}$, the solution is $n= 0$. If $n+ 2 = 2n$, $n = 2$. If $n - 2 = 2n$, $n = -2$. If $\frac{n}{2} = n - 2$, $n = 4$. Finally, if $\frac{n}{2} = n+ 2$, $n = -4$. Thus, there are $\boxed{5}$ such $n$.