A charge $q$ is placed at one corner of a cube as shown in the figure below. The flux of the electrostatic field $\vec E$ through the shaded area is:
I tried this many times. But I am not getting any idea on how to proceed. I tried to find symmetry of flux by imaging 8 cubes but it also didn't work. Both the triangles are at different distance. I also tried to divide field lines inside cube using symmetry which eventually passes through shaded area but not getting it correctly.
Try looking at the problem in this way. The two shaded areas are half triangles. Join them together, and you will get exactly one face of the square. Try looking at the figure till you convince yourself this is true.
You are right to consider 8 cubes. As you know, 8 cubes have 24 outer faces. Joining the two triangles gives you one of the faces. Thus, two half triangles cover the same surface area, as one complete face.
Hence, the flux through one face ( or two triangles ) is $\frac{1}{24}$ of the total flux.
Hope this helps.
Hmm, I'm not actually sure there's total symmetry here. Flux isn't uniform within a face of the cube. If a whole face were shaded, then yeah that would be 1 of 24 surfaces that are perfectly symmetric about $q$ and therefore have 1/24th of the total flux each. But within a given face, some parts are closer to $q$ and thus have more flux per area than other parts of the face.
For that top shaded triangle, it's 1 of 2 halves of a face that are symmetrical about $q$, so it's safe to say it's 1/48.
But the bottom shaded triangle is not symmetrical with the unshaded triangle on the same face: it's farther from $q$ than its unshaded mate. I'm also not seeing any way to resolve it with another argument from symmetry; if you can't see one either, you'll need to actually take the integral. It's a pretty easy one.
Also, side rant: in the real world, what everyone does is take the integral via computer numerically. You don't need fancy simulation software, it's just 2 nested for loops in Python and takes 5 minutes to code and 1ms to run. Most academic programs are way behind in that they act like taking the integral is something to avoid at all costs, just because it can't be done on a blackboard.
