System of 6 harmonic oscillators 
I have seen this question in given  image many times in past few years. And everytime i see this question, i instinctively conclude that its has atleast three possible answers $π/3$ , $2π/3$ and $π$ .
The reason behind my conclusion is that i can compare this system of 6 harmonic oscillations with same frequency and amplitude and different phases, to a system of 6 vectors with equal magnitude originating from same point. To get the zero resultant vector in analogous case of 6 vectors, its very easy to imagine that angle between two consecutive vectors should be $π/3$ ,$2π/3$ or $π$. So in the case of 6 harmonic oscillations the phase difference between two consecutive oscillations should be $π$ , $2π/3$ or $π/3$.
I can mathematically prove this too and calculate its principal solutions.
But for some reason everywhere in saw the solution of this question , the answer given was always $2π$ . Which is a wrong answer according to me. This question was in one my exam too.  And in that exam too the answer in the official  answer key was $2π$ for some reason
Is my reasoning wrong? Or $2π$ is actually the only solution?
 A: Any configuration for which the resulting amplitude equals zero requires the complex sum
\begin{equation}
\sum_{n=0}^5 \exp(\imath\phi_n)=0
\end{equation}
with  imaginary unit $\imath$.
As we have $\phi_{n+1}=\phi_n+\Delta\phi$ for $n\in\{1,2,3,4,5\}$, we obtain
$$\sum_{n=0}^5 \exp(\imath\phi_0) \exp(\imath n \Delta\phi)=0$$
and thus
$$\sum_{n=0}^5 \exp(\imath n \Delta\phi)=0$$
which is satisfied for $\Delta \phi = \pi/3$ but not for the other three options you specify. In particular, as already pointed out by Buzz, $\Delta \phi = 2\pi$ would yield all oscillators with identical phase, so they would have zero phase only at one moment in time and not zero amplitude.
The images below illustrate $\exp(\imath \phi_n)$ with $\phi_n=n\Delta_\phi$ and
$\Delta \phi \in\{\pi/6,\pi/3,\pi/2, 2\pi, 2\pi/3\}$.

Note that for $\Delta \phi =\pi/2$, the two phases at zero and $\pi/2$ occur twice, whereas at $\Delta \phi =3\pi/2$ each phase occurs exactly twice. Thus, out of the given solutions, only $\Delta \phi =\pi/3$ satisfies the condition of zero amplitude. Another independent solution is $\Delta \phi =2\pi/3$. Yet another solution $\Delta \phi =\pi$ has been mentioned by the OP. All of the solutions are trivially extended by adding integer multiples of $2\pi$, e.g.  $\Delta \phi =\pi/3+k\times 2\pi$ with $k\in\mathbb{Z}$.
