## Morris: …and the clocks struck thirteen

“It was a bright, cold day in April, and the clocks were striking thirteen.”

opens George Orwell’s novel ‘Nineteen Eighty-Four’.

1.  By an amazing coincidence thirteen squared is 169 which is the number of times my clock read the right time recently in a single calander day.  Normally it only reads correctly 164 times in a calander day.  This is even more surprising as my clock has been stopped for several years.  How can this be?

My solution combines a number of different techniques.  If you can think of any way a stopped clock can read correctly more than twice a day please post in the comments.  If you can think of something I’ve missed then we may be able to get a bigger answer!

2. I have a second clock which runs slightly fast and I have no way of adjusting it.  How can I make my clock read the right time?

3.  I noticed recently that my third clock was two minutes fast.  It runs at one minute per minute.  It tells me the right time four times a day.  Why?

Thanks to New Scientist’s Feedback Column and it’s readers for some of the idea’s here.

1. ### strauss said,

July 6, 2009 at 4:22 pm

Wow! Great link Steve! Tricky puzzles too…

2. ### hall said,

July 6, 2009 at 4:47 pm

#2 – accelerate it to achieve time dilation…my first thought, maybe to easy…

3. ### Stephen Morris said,

July 6, 2009 at 6:58 pm

Hall,

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?

4. ### Stephen Morris said,

July 17, 2009 at 3:21 pm

Maybe time to post some solutions.

[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]

5. ### Sean said,

August 19, 2009 at 12:10 pm

How about an analog clock with no numbers at the hours?
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.

6. ### Sean said,

August 19, 2009 at 1:21 pm

Another idea is a stopped clock with no numbers or lines whatsoever.  All you have to go on is the angle of the hands.  This clock will be right 11 times per 12 hour period, or 22 times per day.
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.

7. ### solak said,

September 19, 2009 at 5:55 pm

And if the (broken) time mechanism is tiny and the whole thing is a lucite block, then the transparent clock face can be read from either side, so we can double the number of possible readings by using the mirror image.

8. ### Kyle said,

November 17, 2009 at 4:18 am

3: The clock is running backwards and you coincidentally saw it when it was positioned two minutes before the current time.