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### Clock Puzzle

Clock Puzzle - 17 March

How many times in a given day , minutes and hour clock comes in a straight line ?

1. Never if the clock is a defected one or does not h
ave needles..... then also not possible because according to the puzzle both min and hour clock are different....

2. 48 times from 12 to 1 two times i.e)at 12'o clock and approx 12:30 similarly for 24 hours 24x2=48 times

3. 24. There are 24 hours in one day, so the two hands must cover each other once every hour, at some stage.

4. 48 times.

5. Its 23 times
Because during 24 hours of the day, the last hour will not be met for ex:- if we consider 12 am to 12 am, then first time at 12:34 or something but for the last hour (we have already 23 counts of st lines), then we should not count the 12:34 again as it exceeds 24 hours of the day.

6. 120. 60 notches on a sweeping hand clock, telling minutes, will be in sync with hour and minute in line. Another 60 for hour and minute to be overlapped.

7. This comment has been removed by the author.

8. Every minute, the minute hand moves exactly 6 degrees (360 degrees = 60 minutes, 360/60 = 6). In 1 hour, the hour hand will move exactly 30 degrees (360/12 = 30). Therefore, every minute, the hour hand moves 30/60 degrees, or 0.5 degrees, which means overall the angle between the minute hand and the hour hand is 5.5 degrees ahead, each time. Therefore, if each minute there are 5.5 degrees gained between the two hands, then, the number of minutes until they are 180 degrees apart, will be 180 / 5.5, which is 32.7272 minutes, which means the time that the two hands are exactly 180 degrees apart, will be 12:32:44 (to the nearest second). From that point, they will get closer by 5.5 degrees, for another 32 minutes, and 44 seconds, until they are aligned again, which is also a straight line. The question is therefore, basically, how many periods of (32 minutes and 43.63636... seconds) are there in 24 hours? Obviously there are exactly 48 periods of 30 minutes in 24 hours. One way to work this out, is to work it the total number of degrees that the second hand has travelled.

In 32 minutes the minute hand has travelled 192 degrees.
In another 44 seconds, the minute hand has travelled 44/60 * 6 degrees, which is obviously 4.4, or more accurately, the minute hand has travelled 192 + 4.363... or 196.363... degrees. This can be expressed as 196 4/11, or 2160/11 degrees.

There are 1440 minutes (24 * 60) in one day. Each minute is 6 degrees, therefore, 24 hours, means the minute hand has travelled exactly 8640 times. 8640 / (2160/11) = (8640 * 11) / 2160, or 11 * (8640/2160).
8640 / 2160 is exactly 4 (since 8000 / 2000 = 4, 640 / 160 = 4).

Multiply this by 11, and, the answer is exactly 44.

The list of times that the hands are a straight line are (to the nearest second) for the first 6 hours are:
1. 00:00:00
2. 00:32:44
3. 01:05:27
4. 01:38:11
5. 02:10:55
6. 02:43:38
7. 03:16:22
8. 03:49:05
9. 04:21:49
10. 04:54:33
11. 05:27:16
12. 06:00:00

Therefore, every 6 hours, the hands are aligned 11 times. Hence 44 times in 24 hours.

1. *I meant to say "the number of degrees the minute hand has travelled)". Working with the number of degrees the second has travelled would be over the top, as the second hand travels through 1400 revolutions (minutes) per day, which means 1440 * 360, which is far too large for working with. Where as the hour hand would be too small, or full of decimals.

9. Thanks for your explanation of this!