Answer
$\Phi = \frac{1}{24}~\frac{q}{\epsilon} = 0.0417~\frac{q}{\epsilon}$
Work Step by Step
We can use Gauss' law to find the electric flux through the cube:
$\epsilon_0~\Phi = q_{enc}$
$\Phi = \frac{q_{enc}}{\epsilon_0}$
Note that this flux only passes through three faces of the cube. By symmetry, the flux through each of these three faces is $\Phi = \frac{q_{enc}}{3\epsilon_0}$
We could place eight cubes so that a corner of each cube met at the point charge $+q$. Then the charge enclosed by each cube would be $q_{enc} = \frac{+q}{8}$
We can find an expression for the electric flux through each of the other three cube faces:
$\Phi = \frac{q_{enc}}{3\epsilon_0}$
$\Phi = \frac{q/8}{3\epsilon_0}$
$\Phi = \frac{1}{24}~\frac{q}{\epsilon_0} = 0.0417~\frac{q}{\epsilon_0}$
