To solve this problem, I need to recall what complex numbers are and how they relate to the complex plane. A complex number is a number of the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit defined by $i^2=-1$. The complex plane is a coordinate system where the horizontal axis is the real part and the vertical axis is the imaginary part of a complex number. For example, the complex number $3-4i$ corresponds to the point $(3,-4)$ in the complex plane. I can also represent complex numbers using polar coordinates, where I specify the magnitude and the angle of the complex number in the complex plane. The magnitude of a complex number $z=a+bi$ is given by $|z|=\sqrt{a^2+b^2}$, and the angle of $z$ is given by $\theta=\tan^{-1}\left(\frac{b}{a}\right)$, measured in radians from the positive real axis. For example, the complex number $3-4i$ has magnitude $|z|=\sqrt{3^2+4^2}=5$ and angle $\theta=\tan^{-1}\left(\frac{-4}{3}\right)\approx -0.9273$. Using Euler's formula, I can also write any complex number in the form $z=|z|e^{i\theta}$, where $e$ is the base of the natural logarithm and $i$ is the imaginary unit. For example, the complex number $3-4i$ can be written as $z=5e^{-0.9273i}$. This form is useful for multiplying complex numbers, since I can use the property that $e^{a}e^{b}=e^{a+b}$ for any real numbers $a$ and $b$. For example, to multiply $3-4i$ by itself, I can write $(3-4i)^2=(5e^{-0.9273i})^2=5^2e^{-0.9273i-0.9273i}=25e^{-1.8546i}$. Now, I want to find the complex numbers $z$ such that $z^{24}=1$. This means that $z$ is a $24$th root of unity, or a complex number that lies on the unit circle in the complex plane and has an angle that is a multiple of $\frac{2\pi}{24}$. There are $24$ such numbers, and they are given by $z=e^{i\frac{2\pi k}{24}}$ for $k=0,1,2,\dots,23$. For example, $z=e^{i\frac{2\pi}{24}}$ is a $24$th root of unity with an angle of $\frac{\pi}{12}$, and $z=e^{i\frac{2\pi \cdot 10}{24}}$ is another $24$th root of unity with an angle of $\frac{5\pi}{6}$. To find out how many of these $z$ have the property that $z^6$ is a real number, I need to consider what happens when I raise $z$ to the sixth power. Using the property that $e^{a}e^{b}=e^{a+b}$, I can write $z^6=e^{i\frac{2\pi k}{24} \cdot 6}=e^{i\frac{\pi k}{2}}$. For $z^6$ to be a real number, its imaginary part must be zero, which means that $\sin\left(\frac{\pi k}{2}\right)=0$. This happens when $\frac{\pi k}{2}$ is a multiple of $\pi$, or when $k$ is an even integer. Therefore, out of the $24$ values of $k$ that range from $0$ to $23$, only the even ones make $z^6$ a real number. There are $12$ even integers from $0$ to $23$, so there are $12$ complex numbers $z$ such that $z^{24}=1$ and $z^6$ is a real number. 
There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number?