There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number?
Note that these $z$ such that $z^{24}=1$ are $e^{\frac{ni\pi}{12}}$ for integer $0\leq n<24$. So $z^6=e^{\frac{ni\pi}{2}}$ This is real if $\frac{n}{2}\in \mathbb{Z} \Leftrightarrow (n$ is even$)$. Thus, the answer is the number of even $0\leq n<24$ which is $\fbox{12}$.