If the clock’s hands are both the same size, then the clock will be right 44 times per day unless it happened to stop when the hands were directly on top of each other. Then it would only be correct 22 times per day. ]]>

If the hands of the clock tick only 12 times per rotation (i.e., the hour hand ticks once per hour) it would be impossible to tell where 12 is by looking at the location of the hour hand so the clock would be right once an hour or 24 times per day! But I’ve never seen a clock that actually works this way so it seems like cheating.

But I have seen plenty of clocks where the minute and hour hand click 60 times per rotation — so the hour hand ticks only once every 12 minutes. If this clock stops, it is right either four times a day or six times a day depending on where it stops. (This would make for a good puzzle since its probably hard to think of other ways you could make a stopped clock right 6 times a day). If the clock stops in either the first 12 minutes of the hour or the third 12 minutes of the hour it will be right 6 times a day. For example, if the clock stops at 11:00 it could also be rotated to read 12:05 or 1:10. With the a.m. – p.m. thing, this means the clock is right 6 times per day. ]]>

2. is answered in the comments above.

3. This is my answer:

[spoiler]The clock is running ** backwards **at one minute per minute. It is a twelve hour clock. Suppose it is correct at midnight. It will also read the right time at 6am, noon and 6pm. That’s four times a day.[/spoiler]

1. Okay this is the tricky one. You may not like all aspects of my solution. There are other ideas I rejected as being too much like cheating. Make your own minds up as to what is reasonable. I’ve improved my answer a bit since posing the puzzle.

I’m still struggling with the spoiler tag so don’t read the rest if you don’t want to know my answer.

[spoiler]I use three ways of getting a stopped clock to read correctly more than twice a day.

First: it is a digital clock that can be read in a mirror or upside down. So it could read 01:20, 01:50, 02:10 and 05:10. It will read correctly eight times in a day.

Second: You can get on a jet and take a trip through the worlds time zones. According to Windows there are 32 time zones in use, the obvious 24 plus eight others. For example Kabul is GMT +4:30, Kathmandu is GMT + 5:45. If you take your stopped clock with you it will read correclty 32 times in a single calander day.

Third: On the day the clocks go back a stopped clock can read correctly on three occasions.

Putting all this together gives my full solution. I need to make the question more explicit.

‘On how many occasions can a stopped clock read the correct time on a given calander day?’

I have a digital clock which is stopped at 01:20. Using a web-cam and a mirror I create four video streams which show it reading 01:20, 01:50, 02:10 and 05:10. I have agents in every time zone watching my streams and recording when they read the right time.

We have to avoid double counting. For example when it is 01:20 PM in one time zone it may be 01:20 AM in another. This would only count as one occasion. As it happens this problem only occurs for the 24 ‘obvious’ time zones, the other time zones are never twelve hours apart. So each time will be read three times in the twelve ‘paired’ time zones (time zones that are twelve hours apart) and twice in the other eight time zones. That makes 3×12 + 2×8 = 52 occasions.

As there are four times that makes 208 occasions.

When I originally constructed the puzzle I had the clock stopped at a different time which meant I only reached 164 occasions. Can you work out what time I chose?

So what happens on the day the clock goes back? I’m going to make an assumption that the clock goes back from 2AM to 1AM on the same calander day in every time zone. This is a very big assumption, especially as it isn’t true. However it does make it possible to calculate the answer, which is more important.

Again we need to be careful about double counting. When it is 1:20AM for the first time in one time-zone it will be 1:20AM for the second time in the next time zone along. In total we get five extra occasions for 1:20AM and 1:50AM making ten extra occasions. * (It was five extra in total when I originally formulated the puzzle).*

So that makes 208 occasions on a normal day and 218 on the day the clocks go back.

Can anyone improve on this?[/spoiler]

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I would like to live in your world where time dilation is easy.

[spoiler]

You’ve got it right. I was thinking about GPS satellites. They make the clocks run slightly fast to compensate for relativity.

I just put my clock into the right orbit and relativity will make it appear to run at the right speed.

Of course reading it could be tricky, but that’s just an engineering problem.

[/spoiler]

One down, how about the other two?

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