I was given a problem in class. Let's say I bought a clock with no numbers on it (has only the hands). Because of the recent time change, I had to change my clock one hour back. In order to do this, I can rotate the clock.
Is it possible to rotate it exactly one hour?
If every specific time on a clock has a certain distance between the two hands, it cannot be that two subsequent hours have the same distance between the hands.
let $\alpha$ be the angle of the short hand with respect to $12$ o 'clock and $\beta$ the angle of the long hand. Then we know that for clocks the rate of change (the speed at which the hands move is different) of $\beta$ is greater than that of $\alpha$, so consider $\alpha '$ and $\beta'$ to be the angles an hour later, we know that $$ \beta - \alpha \neq \beta' - \alpha'$$
Since the angles between the hands are different there is no way to solve this by rotating the clock, as this preserves the angle between hands, and we need the angle to change.
As a concrete example, suppose it is 12 o 'clock. The two hands must be aligned, an hour later we have that the two hands are no longer aligned, we can not fix this by rotating.
No. If you let the clock run for exactly an hour, the angle between the hands changes (by $30°$ for an ordinary clock).
why is this hard to solve?
just keep rolling the minutes handle backwards until it reaches the same position
edit for super specific people :
put your finger on the tip of the minutes hand, change the minutes hand counter clock wise until it reaches the tip of your finger again
