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.
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.
The clock could have fallen down at
8h24:00, or 3h36:00
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
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
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