Three 24-hour clocks show the time to be 12 noon. one of the clocks is always correct one loses a minute every 24 hours and one gains a minute every 24 hours. how many days will pass before all three clocks show the correct time again?

There is one clock that shows the right time so we do not have to worry about the one which is always correct.

Talking about the second clock that loses a minutes in every 24 hours (or in a day), so after 60 days (since it has lost 60 minutes because it is losing 1 minute everyday) it will show 11:00 a.m when it is exactly the noon.

So this way, in total it will take days before it shows the correct noon.

Now, the third clock gains a minute every 24 hours (or in a day) , after 60 days (when it has gained 60 minutes or a complete hour) it will show 1:00 p.m when it is exactly the noon.

This way, it will take days (since it has gained a minute everyday) when it shows the correct noon.

Therefore, it will take 1440 days before all the three clocks show the correct time again.