Watches in advertisements tend to be set to 10:10 (unless you're Apple). The reason seems to be that the hands are symmetrical when they're at 10:10.
Except they're not. The hour hand moves continuously, not discretely; that is, at 12:30, for instance, the hour hand will not be pointed straight up, but rather halfway between the 12 and the 1.
Well, to make things even more confusing, the minute hand doesn't move discretely, either.
Now, I don't know about you, but I hate things being imprecise. So where should the second hand be such that the hour and minute are symmetrical? And I mean exactly where should it be. If that requires a fraction, or an infinite series, or whatever, so be it. But no rounding.
And just because I'm nitpicky, would you mind working out the same math for all symmetrical pairs, one for each hour? (That is, roughly 1:55, 2:50, 3:45, etc. 12:00 and 6:00 are too easy and are therefore not required for our purposes.)
I'll tell you all in advance: the second hand won't be symmetrical. That just drives me nuts, but I guess I'll have to live with it. Such is life. But please tell me: is there any time for which the three hands are rotated at the same interval around the clock (that is to say, 120º or 2/3πrad between each of the three hands)?
Oh, and since I'm still OCD, would you mind showing your work? You know, I want to make sure it's exact...