[What happened to 1698? It was boring, that’s what happened.]
Spoiler for New Scientist Enigma 1699:
I woke up one morning last summer near the time of the solstice, and noticed that the clock in the darkened bedroom indicated a time either between 3.40 and 3.45am or between 8.15 and 8.20am – I couldn’t tell which.
When I mentioned this to my eccentric uncle, he produced a defunct clock, which he had altered by making the minute and hour hands identical. He then arranged the hands to a particular setting which corresponded to my bedroom dilemma, and told me that it would be impossible to distinguish, by appearance alone, which of two particular times was being shown by the clock.
Assume that at noon both hands on my uncle’s clock point exactly to the 12 marker. Tell me, to the nearest second, what is the difference (less than 4 hours 40 minutes) between the two times which could have been indicated by my uncle’s clock?
One of the hands, call it hand A, is between the 3 and the 4 while the other, B, is between the 8 and the 9. We don’t know which is the hour hand. Let a be the position of hand A as a number between 0 and 60 (so 0 means pointing to 12, 15 means pointing to 3, 30 means pointing to 6, and so on); likewise let b be the position of hand B.
If A is the hour hand then it is fraction b/60 of the way between 3 and 4; that is, a = 15 + (b/60)*5. Likewise if B is the hour hand then b = 40 + (a/60)*5. Or more simply, a = 15 + b/12 and b = 40 + a/12. Solving these we get a = 2640/143 = 18.4615385 = 18 + 27.69/60 and b = 5940/143 = 41.5384615 = 41 + 32.31/60. So the two possible times are 8:18:27.69 and 3:41:32.31 which differ by 4:36:55 to the nearest second.
Which is kind of boring too.