Assume that you have a perfect twelve-hour clock, with an hour hand and minute hand, each of which moves continuously.

At certain times of day, the minute hand and hour hand will be pointed in exactly opposite directions. The obvious case is that of six o'clock. What are the other times? (Specify your answers precisely, rather than rounding to the nearest minute or second.)

At certain times of day, the minute hand and hour hand will be pointed in exactly the same direction. The obvious case is that of twelve o'clock. What are the other times? (Again, specify your answers precisely.)

(The actual mathematics here is very simple. And the insight that would allow one to answer the first question should apply with very minor modification to the second one.)