An interesting problem from Real Analysis-Abbot This is taken from "Undestanding Analysis"- Abbot,
Exercise 4.5.8: Imagine a clock where the hour hand and the minute hand are indistinguishable from each other. Assuming the hands move continuously around the face of the clock, and assuming their positions can be measured with perfect accuracy, is it always possible to determine the time?
From my point of view, I think we CAN tell the time because if you observe for, say, 5 minutes, you can see that the minute hand moves faster than the hour hand. Having that, you can determine which one is the hour hand and which one is the minute hand, and hence, you can easily deduct the exact time. However, this is not very mathematical, and although there is a solution for this question, I do not want to look at it so it would be better that I come up with my own original solution. 
So, please help me, did I answer this question correctly? How do I move up my argument to a solid proof statement? I thank you very much for your help.
 A: Take $[0,1]$ as an interval of 12 hours. Then it is easy to see that at the time $t \in [0,1]$ the hour hand has an angle of $2\pi t$ (in radians), while the minute hand has an angle of $24\pi t$.
So we can consider a model of a clock with the following function
$$f: [0, 1) \longrightarrow S^1 \times S^1$$
$$t \mapsto (e^{2i\pi t}, e^{24i\pi t})$$
If the two hands were distinguishable, then clearly we would be able to say what is the time just as seeing the clock, since the function $f$ is injective.
Now, your problem is equivalent on asking if the following system has a solution
$$\left\{
\begin{matrix}
e^{2i \pi t} = e^{24i \pi s} \\
e^{2i \pi s} = e^{24i \pi t}
\end{matrix}
\right.
$$
where $s \neq t$ are two distinct times in the interval $[0,1)$. This system is equivalent on asking that $t-12s, s-12 t$ are both integers. By trial and error, I found that at least one solution exists, precisely
$$(t,s) = \left( \frac{12}{143} , \frac{1}{143}\right)$$
is the solution of
$$
\left\{
\begin{matrix}
t -12s &=& 0 \\
s - 12 t &=& -1
\end{matrix}
\right.$$
so the answer is that there exist distinct times corresponding to equivalent representation of the clock, however it is easy to see that these are only a finite number.
