## Clock never showing right time might be preferable to the one showing right time twice a day

What clock did you have in mind? If it's a digital watch the problem, I am afraid, does not make a lot of sense. I've been thinking about old-fashioned watch. No functioning watch is absolutely accurate. So one can say no regular watch actually shows absolutely right time. On the other hand, the watch that stopped will show the right time twice a day. Which one would you prefer?

The progress has its downside. Probably in a while not many people will remember or have heard about a regular (what then will be regular?) watch. The late Isaac Isimov once asked How would you then explain what is the positive (counterclockwise) or negative (clockwise) direction?

So while we know what we are talking about, here are a few "regular" clock problems.

1. Besides 12 o'clock, at what other times of day do the hands meet?
2. When are the hands pointing in opposite directions?
3. At what time of day does the line from 6 to 12 bisect the angle between the hands?
4. For a clock, it takes three seconds to strike three. How long will it take for the same clock to strike seven?

[Howard Eves, 76°] elaborates and provides an additional exercise:

There's a story going the rounds about a Dutch professor who fed this question into a very sophisticated computer. "I have the choice between two watches; one is broken and irrevocably stopped, the other loses one second per 24 hours. Which watch should I buy?" The computer's reply: "The one that is stopped, as it indicates the correct time twice every 24 hours; the other does so only once every 120 years."

How did Eves come up with the answer of 120 years?

### Reference

1. H. Eves, Mathematical Circles Revisited, MAA