Clock puzzle
Dudeney's 536-puzzle collection has an interesting problem about an analog clock. (I don't think digital clocks even existed when Dudeney wrote the problem.) Suppose you have a clock with identical hour and minute hands. Some times are unambiguous. For example, if you see one hand pointing directly to the right and one hand pointing directly up, you know it is 3:00; there is no other interpretation possible. (It can't be that the hand pointing straight up is the hour hand, because it would have to be 1/4 past the hour, whereas in that impossible interpretation it is right on the hour.) Dudeney asks what the first time after midnight when an ambiguity occurs will be -- when you can't tell what time it is by looking at the clock.
I haven't solved this yet, but I did discover precisely when all the times are at which the hour and minute hands co-incide. Of course, this happens 13 times from midnight until noon (including both midnight and noon), because the minute hand will have to cross the hour hand once each hour. The first of these times after midnight is 1 hour and 1/11th hour past midnight (around 1:05:45).
