Swapping hands in a generalized clock Consider a generalized clock,
where the minute hand
goes n times as fast as the hour hand,
where n is a positive integer.
The standard clock has
n=12 (sometimes n=24).
As which times can
swapping the hour and minute hands
result in a legal time?
In particular,
for each hour h
from 1 to n,
for which minutes
does this happen?
This obviously happens
when the hands point
in the same or opposite directions.
Are there any other times?
 A: Let the hour hand be pointing at an exact value $H \in [0,1]$ ($0$ represents "12 o'clock" or the angle zero.  $.5$ represents "6 o'clock" or the angle 180 or $\pi$ and $1$ represents 360, "12 o'clock" or $2\pi$ or "full circle"). Then the minute hand must be pointing at a precise value $M = \{nH\}$.  i.e. the fractional part of $nH$ that is $nH = \lfloor nH\rfloor + \{ nH\}$ where $\lfloor nH\rfloor\in \mathbb Z$ ad $0 \le \{nH\} < 1$.
That's what a legitimate time is:  if $M = \{nH\}$ the time is legitimate.
So we need times where $M = \{nH\}$ and $H = \{nM\}$ or $H = \{n\{nH\}\}$
$\{nH\} = nH - \lfloor nH\rfloor$; $n\{nH\} = n^2H - n\lfloor nH\rfloor=\{n^2H\} + \lfloor n^2H\rfloor- n\lfloor nH\rfloor$. $0 \le \{n^2H\} < 1$ and $\lfloor n^2H\rfloor- n\lfloor nH\rfloor\in \mathbb Z$.
So $\{n\{nH\}\} = \{n^2H\}$.
If $H = \{n^2H\}$ then  $H = n^2H - k; H = \frac {k}{n^2-1} \in \mathbb Q$.
And, that's that.  If $H = \frac {k}{n^2 - 1}$ then $H$ is a "reversible time".
Example:  for the 12 hour clock, there are $143$ of these times.  (Which actually surprises me as I assumed there were only 11.) 
