## The Clock on the Mantlepiece

Posted by Karl Sharman on July 1, 2011 – 6:35 am

I visited a friend one Sunday (I have friends – shocking, I know). I noted something unusual about his clock, other than the fact it was analogue, when the radio gave the time at the hour, the analogue antique was exactly 3 minutes slow.

“It loses 7 minutes every hour”, he told me, “no more and no less, but I’ve gotten used to it that way.”

When I spent a second evening with him later that same month, I remarked on the fact that the clock was dead right by radio time at the hour.

It was rather late in the evening, but he assured me that his treasure had not been adjusted nor fixed since my last visit.

What day of the week was the second visit?

July 1st, 2011 at 7:05 am

Maybe I’m messing this up.

Seems like that Tuesday evening, the clock would be on the hour, but it would actually be 3 hours later in the evening than the Sunday visit.

The original text does state ‘it was rather late in the evening’.

We can see this 3-hours-later but clock showing on the hour for other days:

– The next Sunday (start of week 2)

– Friday of week 2

– Wednesday of week 3

– Monday of week 4

– Saturday of week 4

– Thursday of week 5 (but this is definitely into the next month)

July 1st, 2011 at 10:32 am

I might be overlooking something here. Assuming you are looking at a 12 hour clock, it would be 717 minutes fast ((12 * 60) – 3). Losing 7 minutes/hour it would take 102.4286 hours or 4.268 days to slow down to the exact time. After that it would be right every 102.857 hours or 4.286 days. I don’t see a case where it would be correct on the hour within a one month period.

July 1st, 2011 at 10:50 am

You went back 16 days and 3 hours later on Tuesday. That is when the clock will read the correct time.

July 1st, 2011 at 11:58 am

Because it’s an analog clock, it only needs to lose 12 hours.

July 1st, 2011 at 10:44 pm

17 days and 3 hours later, on a Wednesday.

Clock loses 7 minutes each hour after being 3 minutes slow. So after 51 hours (on the radio) it has lost 3+7*51 = 360 minutes, i.e 6 hours slow after 2 days and 3 hours following the Sunday visit.

Thereafter the clock loses 7 hours every 60 hours of radio time. So it will lose another 42 hours in a further 6*60 hours or 15 days of radio time.

Therefore, in total, it will have lost 47 hours and 57 minutes after 17 days and 3 hours but will show the apparently correct time again on the third Wednesday after the initial Sunday visit..

July 2nd, 2011 at 12:17 am

Tuesday?

Since the clock loses 7*24=168 minutes a day, and since it was already three minutes slow, it would only have to go back 1440(minutes in a day)-3=1437 minutes. In 8.553571429… days, it will have lost 1437 minutes, so then the day of the week will be Tuesday.

July 3rd, 2011 at 2:10 am

Some excellent answers, with some great explanations, mostly concurring that Tuesday is the day. Well done all who got Tuesday, except it’s not Tuesday….. ;-(

Don’t forget it is an analogue clock, but in the evening – might as well be a 24hr clock….

Meantime, Cazayoux is hedging his bets in post 1….

July 3rd, 2011 at 3:21 pm

I’m guessing that there is more than one possible answer.

Friday of that same week, Wednesday of the week after, and five days following that day.

It takes the clock 60 hours to get back on time with the correct hour. Assuming he visits the same time on both days, the clock will have done this twice by then.

July 3rd, 2011 at 9:04 pm

If the first visit was at 19:00 then the next will have been 12 days later (Friday) at 22:00

If the 1st visit was at 21:00, then next will be 8 days later @ midnight, which is probably no longer considered evening.

July 4th, 2011 at 1:01 am

after 27 days and 4 hours. 4th saturday after that sunday.

July 4th, 2011 at 7:16 am

if the first visit was on sunday on week 1 of the month,

the next visit will be on wednesday on week 3 of the same month 3 hours later of the first visit,

as the analogue clock will give the same time every 411 hours. DON’T forget the first 3 minutes.

July 4th, 2011 at 9:16 am

Need to know what month you first went to your friends.. otherwise DST could be in play during march.

February has an extra day during a leap year too.

July 4th, 2011 at 6:11 pm

Let’s look at this another way . . .

The clock is 3 minutes slow on the first Sunday visit and loses another 7 minutes each passing hour. Thus it will be 10 minutes slow after one hour, 17 minutes slow after two hours, and so on.. To come close to showing the correct time the clock now has to lose 12 hours less 3 minutes, i.e. 717 minutes. But 7 doesn’t divide exactly into 717, so the clock will be 3 minutes fast and then 4 minutes slow on either side of the correct time after 12 hours.

So instead of 12 hours we must try 24, 36, 48, etc hours until we get the exact correct time. This means successively adding 720 minutes to our initial 717 minutes until we get an exact multiple of 7. This happens when the clock has lost 2877 minutes. At 7 minutes per hour this means that the total time lost is 2877/7 = 411 hours.

So 411 hours have elapsed between visits. This comes to 17 days and 3 hours, which takes us to three hours later on the third Wednesday after the initial Sunday visit, as predicted in post # 5 above.

July 5th, 2011 at 1:39 pm

The answer is 17 days and 3 hours later, which would have been a Wednesday. This is the only other time in the same month when the two would agree at all.

Wizard of Oz wins the “Telling the Time” badge.

In 17 days the slow clock loses 17*24*7 minutes = 2856 minutes,

or 47 hours and 36 minutes. In 3 hours more it loses 21 minutes, so

it has lost a total of 47 hours and 57 minutes. Modulo 12 hours, it

has *gained* 3 minutes so as to make up the 3 minutes it was slow on

Sunday. It is now (fortnight plus 3 days) exactly accurate.

tom queried DST, so, since the clock was not adjusted since the last visit, it’s also possible that the radio time shifted by one hour due to a change to or from summer daylight saving time. However, it turns out that the only additional possibilities that need to be considered are those of 4 days 15 hours later, when the clock would have lost 12 hours 57 minutes, and 29 days 15 hours later, when the clock would have lost 3 days 10 hours 57 minutes. Without even considering the rules for when in the month the clock is changed, these possible solutions are ruled out because we know that both visits were in the evening (“I spent a second evening with him”). and they involve times in a different part of the day.

July 5th, 2011 at 3:27 pm

I re-ran my calcs. Karl is right…(and Wiz)

on the 17th day AFTER Sunday (or 18th day) plus 3 hours, the clock will have lost enough time to be 3 minutes fast had it not been for those 3 minutes it was slow on Sunday.

What gets me is how your friend has “gotten used to it that way”. It is a very rare occurance that the clocks will actually read the same value. In fact, I’ve calculated out past a year from that point (excluding any daylight savings time) and can’t find another time the two read the same. Did he say when the last time the two were, as you put it “exactly accurate”?

July 5th, 2011 at 5:16 pm

Sorry, but I have to disagree.

You seem to be introducing the modulo 12 against the wrong time.

My proof … if you keep subtracting 7 minutes from the time every hour, then you finally hit an ‘hour’ after 51 hours starting from 3 mins early … , 7 mins early, 10 mins, 17, 24, 31, 38, 45, 52, 59, 6, 13, 20, 27, 34, 41, 48, 55, 2, 9, 16, etc until to have 0 mins ‘early’ 51 iterations after the start, when the clock will read 2, and the real time will read 10pm – these hours don’t match so you keep going.

The next matches will be at 111 hours, 171 hours, 231 hours, and 291 hours after the first start. The 291 is the magic one where the clock and the real time will both match at an hour 3 hours later than the first. The point to note is that the REAL time hits a match every 12 hours, starting 3 hours from the initial meeting so the modulo 12 should be used against the real time 3 hours from the start, not the time of the clock.

July 5th, 2011 at 5:19 pm

Sorry, those times are based on a 19:00 start. The modulo 12 is used on the hours -3 (51-3, 111-3, 171-3 etc)

Hope that makes sense.

July 5th, 2011 at 5:21 pm

Oh, and it’s not every 12 hours, it’s at 12 hourly times (10pm, 10am, 10pm etc) with a number of days difference … I seem to not be explaining things very well.

July 6th, 2011 at 7:34 am

Ok, that wrong modulo theory is rubbish, sorry about that. Where I think things differ is that you have generalised and come up with an average maybe ? But this overlooks the fact that it needs to be both in the evening. If the start time varies then so does the length of time it takes to meet the criteria. I’ll investigate more later today.

July 6th, 2011 at 7:38 am

Eketahuna, feel free to ignore my maths, as I think Wizard of Oz has explained it equally well from 2 different angles. Take a look at his posts – hope you will agree by this evening!!

July 6th, 2011 at 2:18 pm

Karl, I’m not saying your maths is wrong, per se, but I think I have worked out what I am trying to say … you seem to be forgetting that the clock is not just losing minutes, it’s losing hours as well … so when the hours align, it’s not a multiple of 12 hours. You don’t seem to have factored that in to the equation ?

July 6th, 2011 at 2:53 pm

Try this … after 12 days … 12 x 24 x 7 = 2016 minutes lost = 33 hours and 36 minutes. After another 3 hours, it’s lost another 21 minutes, so it’s 33 hours and 57 minutes lost, plus the original 3 minutes it was behind gives us an even number of hours lost. 33 hours + 3 hours = 36 hours, your multiple of 12.

July 6th, 2011 at 10:44 pm

Hi Eke – Where do the extra 3 hours in your last line in post 22 come from? From what you’ve written I get 33 hours + 57 mins + 3 mins = 34 hours, not 36. The clock would then appear to be 2 hours fast (or 10 hours slow).

July 7th, 2011 at 1:45 pm

Hi Wiz, you are right, but I didn’t explain myself very well, yet again – sorry about that.

The last line isn’t relevant (it’s wrong, but that’s not important ) as it doesn’t actually have to be a multiple of 12 hours, it just has to have lost a multiple of 60 minutes, so 34 hours is fine (and as you say, the correct answer). If the original time was 19:00 then 34 hours lost from there gives us a time of 10:00 on the analogue clock – so we do a revisit at 10pm and the times match.

July 7th, 2011 at 6:57 pm

Dóh ! 19:00 – 34 isn’t 10:00

July 13th, 2011 at 11:19 am

Saturday?