Can we identify the time if we know every angle between three hands of a watch? Let $M, H, S$ be the minute hand, the hour hand, the second hand of a watch respectively. Also, let $A_{MH}, A_{MS}, A_{HS}$ be the angle between $M$ and $H$, $M$ and $S$, $H$ and $S$ respectively.
Then, here is my question.
Question : Can we identify the time of the watch if we know all of $A_{MH}, A_{MS}, A_{HS}$ of a watch? If we can, how can we identify the time?
Motivation : I came up with this question while I was repairing my watch:)  The answer seems yes, but I'm facing difficulty. Can anyone help? 
 A: The answer is yes. 
If all three hands move continuously, then the angle of the hour hand is enough to completely determine the time, and therefore the angle of the other two hands.
Let $X$ be the angle of the hour hand from the $12$ o'clock position.  Let $Y$ be the angle between the hour and minute hand.  Let $Z$ be the angle between the minute and second hand.
The hour hand moves at a rate of $30$ degrees per hour, while the minute hand moves at a rate of $360$ degrees per hour.  It we use a different frame of reference, it is possible to see that the minute hand moves at a rate of $330$ degrees per hour relative to the hour hand.  This means that the value of $Y$ is equivalent every $360/330=12/11$ hours.
The second hand moves at a rate of $21600$ degrees per hour.  Relative to the $360$ degrees per hour of the minute hand, it is traveling $21240$ degrees per hour.  The value of $Z$ repeats every $360/21240=1/59$ hours.
Now, we need to find a number that is the smallest multiple of $12/11$ and $1/59$.
$$11*59 = 649$$
$$12/11=708/649$$
$$6/59=66/649$$
$$\text{LCM}(708,66)=7788$$
$$7788/649 = 12$$
As we can see, the unique combination of angle $Y$ and $Z$ only repeats every twelve hours.
A: HINT: The time has the same angles for 12:20:40, 4:40:00, etc.
