help! my brain can't function after doing too much of differential equations... anyway, if a clock is set at 3 o'clock exactly, how long does it take for the hour,minute and second hand to be all pointing in the exact direction?
If he says "exactly," he means "exactly." At 3:16:16 the three hands are not exactly aligned, and there is no moment just a bit earlier or later when they are. The clock passes completely through all hours but one without all three hands lining up. Set up the equations for yourself and you'll see that there's only one solution: 12:00:00, which occurs only twice every 24 hours. This is the only spot on the clock that will ever be occupied by all three hands at once. There is no mathematical reason to assume that it will occur any more often than that, because of the ratios of the three hands' speeds. So starting at three o'clock, we have to wait nine hours for the alignment to occur.
