Clock arm perfect alignment problem On a typical round $12$ hour clock with $3$ arms (hour, minute and second), at precisely what time will all $3$ arms be perfectly aligned on top of each other, with the exception of at $00$:$00$:$00$ hours?
High precision is of the essence in this problem. If we consider a time at $3$:$15$:$15$ the arms are not perfectly aligned because the minute clock would have moved forward from the "$15$" position by $\frac{15}{60}$ of a minute etc.
In this problem we assume that the arms move smoothly, rather than in 1 second increments. Any fraction of a second/minute/hour is possible.
I don't know if an answer exists to this problem, but a mathematical computation of a specific time or a proof that no answer exists is what I am looking for.
 A: Suppose that this happens at exactly $x$ hours after $00{:}00{:}00$.  At this time,


*

*the hour hand has travelled through an angle $x/12$ of a full circle;

*the minute hand has travelled through an angle $x$ of a full circle;

*the second hand has travelled through an angle $60x$ of a full circle.


If the hands are to lie on top of one another, these angles must all differ by an integer.  So we have
$$\frac{11x}{12}=m\ ,\quad 59x=n$$
where $m$ and $n$ are integers.  Eliminating $x$ gives
$$12\times59m=11n\ .$$
Therefore $m$ must be an integer multiple of $11$, so $x$ is an integer multiple of $12$.  Therefore the time is a multiple of $12$ hours after the $00{:}00{:}00$ position, and so the hands are still at the $00{:}00{:}00$ position.
That is, the three hands can never be exactly together, except at $00{:}00{:}00$. 
A: I have played around with the equations of clock hands before, but without the seconds hand. What I can contribute is an equation for the angle of the hour and minute hand.
For the minute hand the angle from the twelve o'clock position is $6m$ where $m$ is the minute.
For the hour hand is half the minutes past twelve which is $${60h+m} \over {2}$$
where $h$ is hours. To find where they are equal, set the equations equal, however you need an expression for the angle of the seconds hand.
Edit
The result you get from setting them equal is
$$m=\frac{60}{11}h=h\cdot5.\overline{45}$$
so you can plug in hours ($h$) from 0 to 11.
So you don't need an expression for the seconds hand after all.
These answers will have repeating decimals in them, but you have stated that any fraction of a second can be used.
