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When I came back from my vacation I saw my wall clock has fallen down and not working. The clock was on the floor and all its numbers were detached from their positions. Can you help me to find out at what possible time(s) the clock could have fallen down? The three hands (hour, minute and second) of this clock are exactly the same shape and size. enter image description here

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  • $\begingroup$ Am I correct in that there are 60 tick marks around the outside? I counted 15 in approximately a quarter section but it's easy to be wrong that way. $\endgroup$ – Engineer Toast Oct 18 '17 at 20:10
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    $\begingroup$ @EngineerToast, you are correct, there are 60 tick marks around the clock. $\endgroup$ – Seyed Oct 18 '17 at 20:12
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    $\begingroup$ My intuition tells me to turn the clock around and look for the nail loop.... but that kind of defeats the purpose. $\endgroup$ – John Eisbrener Oct 18 '17 at 20:23
  • $\begingroup$ If the clock were working fine, how would you come to know what time it is (at minute precision), anyway with such a strange clock? $\endgroup$ – Mea Culpa Nay Oct 19 '17 at 7:06
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    $\begingroup$ Does the minute hand move partially on every second? And the Hour hand move partially on every minute (or even second) ? $\endgroup$ – nl-x Oct 19 '17 at 9:57
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Knowing that every hand lies exactly on a mark, the second hand must be pointing towards 12, else the minute-hand would be between marks, and the minute-hand must be pointing at a multiple of 12 marks away from it (12 mins, 24 mins, 36 mins or 48 mins ) else the hour-hand would be between marks.

This leads to two (four, counting for AM and PM per day) possible answers.

If we take the left-most hand in the picture to be the second hand, pointing at 12, and the next hand clockwise being the hour hand, with the final one being the minute hand, we are given 3:36 (or 15:36), which is a valid time for those hand positions.

If we take the rightmost hand shown as the second hand pointing at 12, and moving clockwise, we reach the second hand before the topmost hand being the hour hand, we reach 8:24 (Or 20:24), which is also valid with those hand positions.

Nothing can be made if we take the "middle" hand pointing closest to the top to be the second hand, as neither of the other hands are a multiple of 12 marks away from it, meaning there is no valid position of the minute hand.

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    $\begingroup$ I suppose it could have been one of those clocks where the minute hand jumps from one mark to the next rather than progressing continuously.... $\endgroup$ – David Z Oct 19 '17 at 6:36
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The clock could have fallen down at

8h24:00, or 3h36:00

because

all 3 hands are precisely on a mark. It means that the seconds are necessarily on the 12. And because there are 5 marks per hour, the hours will be on a mark at minute: 0, 12, 24, 36 and 48 of the hour

from there:

when counting the marks the difference between hands is 18, 24, and 18. Those 3 numbers leave only two possibilities: either we are at minute 24 or 36. If we consider 24, then the hour hand is at mark 42 (18+24). And 42/5 = 8 + 2/5 which is fine because 24 is two 5th after the hour. If we consider 36, then the hour is on the 18th mark. And 18/5 = 15 + 3/5 which is fine because 36 minutes is the 3rd fifth of the hour

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    $\begingroup$ How can you tell that all hands are exactly on a mark? One second difference would rotate the minute arm by $\frac{1}{60}$ of the distance between two marks – that's $0.1°$. $\endgroup$ – A. P. Oct 18 '17 at 20:32
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    $\begingroup$ I have a retina display $\endgroup$ – sousben Oct 18 '17 at 20:33
  • $\begingroup$ And you did it well, thanks for you participation. I am sure you will solve my next puzzle and I will give you a big $√$ as a correct answer :-) $\endgroup$ – Seyed Oct 18 '17 at 20:45
  • $\begingroup$ You left out the AM/PM part. There could have been 4 different times. $\endgroup$ – nl-x Oct 19 '17 at 10:18
  • $\begingroup$ @nl-x Given that they came back from vacation, it could have been over several days, so the clock could have fallen down four times a day while they were away. I'm not sure, then, how knowing the time of day it fell down would be useful. The reality, though, is that you could turn it over and find out how it hung to find the "up" side, or inspect the mechanism, and narrow it down to two times. This, of course, also requires that the hands, big as they are, didn't slip and move in the fall. If we're going to be pedantic, that is. $\endgroup$ – Adam Davis Oct 19 '17 at 17:00
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since they (the hands) are divided equally around the clock, we take the original 12 numbers and divide them by 3. This leave 0 4 8. any derivation of these numbers could work... ie. 1 5 9, 2 6 10, 3 7 11 and back to 4 8 12.

Each of these allotments

have 6 potential times per group.

They are as follows configure for both am/pm

>!    1 5 9 => 1:25:45, 1:45:25, 5:10:45, 5:45:10, 9:10:45, 9:45:10
>!    2 6 10=> 2:30:50, 2:50:30, 6:10:50, 6:50:10, 10:30:10, 10:10:30
>!    3 7 11=> 3:35:55, 3:55:35, 7:15:55, 7:55:15, 11:15:35, 11:35:15
>!    4 8 12=> 4:40:00, 4:00:40, 8:00:20, 8:20:00, 12:40:20, 12:20:40
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    $\begingroup$ Hands are not divided equally. $\endgroup$ – Seyed Oct 18 '17 at 20:25
  • $\begingroup$ ah @Seyed i did not catch that! i thought i counted correctly... small screen on the phone. now that i am on a pc i see it. $\endgroup$ – Jason V Oct 19 '17 at 15:49
  • $\begingroup$ No problem, I hope you catch my next puzzle :-) $\endgroup$ – Seyed Oct 19 '17 at 20:27

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