by Howard Williams
Published Sunday May 08 2022
I have an analogue wall clock with a second hand and also a separate 24-hour hh:mm:ss digital clock. The wall clock loses a whole number of seconds over a two-digit period of seconds. The digital clock gains at a rate 2½% greater than the wall clock loses. After resetting both clocks to the correct time, I noticed that they both displayed the same but wrong time later in the same week at a time of day one hour earlier than that when they were reset.
I can reset one of the clocks at an exact hour so that it will show the correct time when the televised rugby kicks off at 19:15:00 on the 31st.
What is the latest time (hour and date) when I can do this?
Sunday Times Teasers Discussion Group 
Makes you wonder why people don’t buy reliable clocks.
Or is that against the spirit of the thing?
😉
Yes, I have an antique analogue clock (wooden pendulum) that manages <1min error per day (better than 0.1%). I don’t understand the digital clock being so bad – I guess we must be back in teaser land again!
My grandfather had a clock, which he wound and set every day. He would say: “Sometimes I think I’m here for the clock’s benefit rather than the other way round…”
I think that the wording of the sentence: “After resetting both clocks to the correct time, I noticed that they both displayed the same but wrong time later in the same week, and one hour earlier than the time of setting.” can be interpreted in more than one way.
I took it to mean that when they both displayed the same wrong time, the displayed time was one hour earlier than the setting time. I.e. I assumed that the sentence referred to what was displayed at that time. The other interpretation is that it was the actual time when the two same wrong times were displayed. If this were the case I would have preferred or expected the wording to be “… week, and at one hour earlier than the time of setting.”
Suffice to say, Brian and I have different answers. If my interpretation is correct, this is quite a challenging teaser, and one does not know when one has the right answer.
The second interpretation has a very small solution space, and I quickly found Brian’s answer.
I am relieved to hear from John Owen that I don’t have more work to do! I have amended the teaser text accordingly.
I found 3 different answers to the puzzle with the interpretation suggested by John.
So I think the intended interpretation is that the actual time is one hour earlier that the time of the setting, not the displayed time.
(Although the setter must have a much more accurate clock than the two mentioned in the puzzle text in order to determine the actual time).
In such a strange collection the wall clock might have been of the rare 24 hour variety.
I started with making expressions for time displayed by the two clocks in terms of correct time elapsed in seconds and their lose/gain rate. An expression for the interval in correct time (in terms of lose/gain rate) for when the two clocks display equal figures, follows easily. Chasing lose/gain ratios that yield whole numbers of seconds for both the two clocks and correct time, results in several lose/gain rate candidates, and, in my case, that goes for both of the two interpretations mentioned by John above..
Every new attack only brings distorted clocks à la Dali to mind. I need a break.
Just for the record (after having read Brian’s uploaded solution): In addition to Brian’s value for s/p, I had 32/63 and 32/93. I did not find the way he eliminates them in line (3) and (4).
I have uploaded a manual solution for this teaser to the password protected area. I would appreciate any constructive feedback (and proof reading!).
Brian
Your equation (5) actually shows that s is a multiple of 32 and that d=4.
Thanks Tony, I’ll amend it.
But 95 is not a divisor of 160.
It seems clearer to me to write the equation as 27s(24d – 1) = 160kp.
Yes, thanks again, loose wording on my part (I want to keep the s/p form though as it is the focus of what follows). I really appreciate the effort you have made to review and improve my solution.
Please note that I have added a NEWS menu item which I hope regulars will look at occasionally.
I failed with this one.
At first I went down the wrong route thinking that it was the displayed time that was 1 hour earlier, later in the week. I was generating too many possibilities so I went back to re-read the question and decided it meant the real time was 1 hour earlier, later in the week, when the clocks read the same wrong time.
This was then easier!
So then I solved the problem, I thought. But I came here to check the answer (not the solution) and I had it wrong.
I did not read the solution before I had had another few goes, leaving the teaser alone for some time in between. But I kept getting the same wrong answer.
So I came back and read the solution. I was very pleased it was available. Thanks Brian. I understand the solution and used the same method for the first part.
At least I got part of the question right: the first bit which allows us to work out the loss and gain rates of the two clocks.
My mistake then was to misinterpret the next bit of the question. I read it to mean ‘SET one of the clocks TO an exact hour’ – not necessarily the right hour! But what it actually says is ‘RESET one of the clocks AT an exact hour’ – meaning the correct time.
My wrong answer was to set the analogue clock (to 03: 00:00) at an actual time of 05:00:00 on 31st!
Moral of the story is I need to be more careful in reading the question.
Can somebody please explain to me how the digital clock knows the reading from the (analogue) wall clock?