The 3 clocks problem
There are 3 clocks A, B & C. They start donging together at midnight and do so every hour. Gradually, B starts taking 2 minutes more than A per hour & C starts taking 2 minutes less than A per hour. At what exact time will A, B & C dong together again?
The answer is 11.
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1 solution
This problem is only for ten points. and surely after 30 hrs the clocks will dong together. And @ Bhaskar Sukulbraham - taking LCM of 58, 60 & 62 is a somewhat wrong method because you have assumed that the 3 clocks lag or exceed their donging by 2 minutes only once. But here is the case that it goes on lagging by 2 minutes every hour. So the clocks should dang after 30 hrs.. The answer should be 6.
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So you want to say that the clock C will dong after 58 minutes (00:58 am), then 56 (01:54 am), then 54 (02:48 am) and so on, and clock B after 62 (01:02 am), then 64 (02:06 am), then 66 (03:12 am) and so on ??
Here's explanation of my method:
Dong | Clock A | Clock B | Clock C
1st time | 00:00 am | 00:00 am | 00:00 am
2nd time | 01:00 am | 01:02 am | 00:58 am
3rd time | 02:00 am | 02:04 am | 01:56 am
4th time | 03:00 am | 03:06 am | 02:54 am
... and so on.
30th time | 05:00 am | 05:58 am | 04:02 am
31st time | 06:00 am | 07:00 am | 05:00 am
32nd time | 07:00 am | 08:02 am | 05:58 am
.
.
871st time | 06:00 am | 1 1 : 0 0 a m | ---
900th time | 1 1 : 0 0 a m | ----------- | -----
931st time | 06:00 pm | ----------- | 1 1 : 0 0 a m
The time shown in above table is the real time, i.e., clock B shows 01:00 am when the real time is 01:02 am (2nd row) and clock C shows 01:00 am when the actual time is 00:58 am, right?
As you can see from above table, they don't dong together after 30 hours , i.e. at 06:00 am. At 05:58 am, Clock B (30th time, clock shows 05:00 am) and C (32nd time, clock shows 07:00 am) will dong together. At 07:00 am, Clock A (32nd time, clock shows 07:00 am) and B (31st time, clock shows 06:00 am) will dong together.
After 37 days at 11:00 am (the answer), all the 3 clocks dong together again, but they don't show the same time.
Clock A dongs for the 900th time (shows 11:00 am of 38th day),
Clock B dongs for the 871st time (shows 06:00 am of 37th day),
Clock C dongs for the 931st time (shows 06:00 pm of 39th day).
Therefore if you want to find the time at which all the 3 clocks dong together again then you have to find LCM of 60, 62, 58.
Now I feel like I deserve more than 10 points. :P
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I'm afraid I think that you're still off. The LCM doesn't help here. In your working you've included the 31st time that they dong, which is also the first time they dong together. Why is this not the answer?
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@Ben Merrett – When the clocks dong 31st time, all the 3 clocks show 06:00am but they dong at 06:00am (A), 07:00am (B), 05:00am (C) respectively.
The time in the table is the actual time, not the time that the clocks show.
For them to dong together, the actual time must be same.
If they were to lag or exceed their donging only once they would never dong together again?
I'm afraid I have to disagree. Surely after 30 hours the clocks will dong again? There is no need for them all to read the same hour in order to dong together, just for them all to be on the hour. Perhaps what you meant to ask was what time they shall read the next time that they all read the same time? You've also overlooked that most donging clocks have 12 -hr faces.
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You're right! Blame the one who asked the question. :P
(sigh) I got the correct answer in my first try using the above method, so I didn't give too much thought since it was only for 10 point.
What do you say? @Nayanmoni Baishya
And for the last part that you say, I don't think the answer depends on the number of hours on the clock face.
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Well it was unnecessary to prove that it was wrong in two ways, but if he had intended it to mean when do the clocks next read the same time I wanted to highlight that he would need to consider the fact that they didn't need to align by 24 hours, only 12. I hope that makes sense.
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@Ben Merrett – Check out my detailed explanation of the method in my reply to @Rishabh Bagawade and let me know if it has any error.
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@Bhaskar Sukulbrahman – @Ben Merrett & @Bhaskar Sukulbrahman Guys lets end up the discussion on this problem.. I'll try to summarize the discussion here.. @ Bhaskar - You have done it correctly.. You have included the condition of lagging or exceeding per hour.. & yes @ Ben - they won't dong together if they lag or exceed only once.. But @ Bhaskar - You have calculated the next time all the 3 will show the same time but in actual the clocks dong every hour so answer can be 30 hrs also.. The only fact here is that the question should have mentioned everything properly.. And yes Bhaskar you deserve more than 10 points..!!!
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@Rishabh Bagawade – Answer cannot be 30 hrs (actual/real 30 hrs, not 30 hrs of clock B or C), since after actual 30 hrs., only clock A dongs. Well Clocks B and C will also dong but after their 30 hrs., not after the actual 30 hrs.
This question confused me in the beginning , but now I think that I'm not wrong.
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@Bhaskar Sukulbrahman – My thinking was that after 30 hrs C will exceed A by exactly one hour and B will lag by exactly one hour.. This will give us a situation when 3 of them will dong together but time will differ by one hour in each of them..
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@Rishabh Bagawade – I am of opinion that the question only asks WHEN they will dong together again - not when they will dong the SAME number of times. As per Bhaskar's table, Dong | Clock A | Clock B | Clock C 1st time | 00:00 am | 00:00 am | 00:00 am 2nd time | 01:00 am | 01:02 am | 00:58 am 3rd time | 02:00 am | 02:04 am | 01:56 am 4th time | 03:00 am | 03:06 am | 02:54 am ... and so on. 30th time | 05:00 am | 05:58 am | 04:02 am 31st time | 06:00 am | 07:00 am | 05:00 am
'2nd time' here means 1 hr after 00.00 am. Therefore, '31st time' only means 30 hrs after 00. 00 am.' which is obviously 6 am (after 30 hours). I stick to my answer of 6 am.
LCM (58, 60, 62) = 53940, is the number of minutes after which the 3 clocks dong together again.
6 0 5 3 9 4 0 = 899, is no. of hrs.
2 4 8 9 9 = 37 2 4 1 1 , is no. of days.
So, after 37 days at 1 1 am, the 3 clocks will dong together again.