Another way of looking at it, is to say that the position repeats after 12 hours. In those 12 hours, the hour hand goes round once and the minute hand twelve times, so the minute hand overtakes the hour hand exactly eleven times. Because the speeds are constant, those moments must be exactly one-eleventh of a 12-hour period apart.

Similarly, the minute hand goes round 12 times and the second hand 720 times. So the second hand overtakes the minute hand 708 times, exactly 1/708th of a 12-hour period apart.

The three hands exactly align whenever the minute hand overtakes the hour hand at the exact moment that the second hand is overtaking the minute hand. That time would have to be an exact number of elevenths and an exact number of 708ths. But as 11 and 708 are coprime, there is no such fraction: the only answer is zero elevenths, which equals zero 708ths, which is 12:00:00.

Thank you, coprime numbers.

Peter