You have an analogue clock with a second hand.


This challenge is one of the classic task to solve questions at Microsoft interviews. Applicants are asked the question how many times a day the hour and minute hands meet each other. As this interview challenge has now become widely known, companies started using its modifications.


Initially, let’s consider the variant of the most expected mathematical solution. First, imagine the situation when the hour and minute hands overlap. Everyone understands that this appears at midnight, then at approximately 1:05, 2:10, 3:15, and so on. Simply put, the hands overlap each other every hour, except for the time from 11:00 to 12:00. At 11:00, the faster minute hand is at 12, and the slower hour hand is at 11:00. They will not meet each other until 12:00 pm, and therefore there will be no overlap in the area of ​​11:00.


Consequently, for each 12-hour period, 11 overlaps happen. They are equally divided in the time since both hands run at a constant speed. Aforementioned means that the periods between overlays are 12/11 of an hour. This is equal to 1 hour 5 minutes 27 and 3/11 seconds. Accordingly, for each 12-hour overlay cycle, overlaps happen through the periods displayed in the image below.

Now, let’s turn back to the second hand. Its overlap on the minute hand is possible when the number of minutes corresponds with the number of seconds. An accurate overlap appears at 00:00:00. Generally, minute and second hands overlap only for a split second. For example, at 12:37:37 PM, the second hand will show at 37, lagging behind the minute one, which at this time will be between 37 and 38 and lag behind the hourly one. Immediately, the minute and second hands will overlap, but the hour hand will not be near them. It means that there will be no overlap of all three hands.


The second hand will not overlap in any of the options in the picture, except for midnight and noon. This indicates that the concluding solution is twice a day.


Now, let’s have a look at the solution which is desirable by Google. The second hand is designed to show small time intervals, but not to show the time with pinpoint accuracy. In case it is not synchronized with the other two hands, this is quite common. The term “synchronization” here means that at midnight and noon all three hands point exactly to 12. Most analogue clocks of all brands do not enable you to set the second hand accurately. 


You will need to remove the battery or wait, in case you have the mechanical clock when the spring factory ends. Afterwards, once the second hand is stopped, synchronize the minute and hour hands with each other, then wait until the clock time comes to return battery or wind up the watch.


In order to carry it out, you need to be a madman or a devotee of punctuality. Otherwise, the second hand will not show “real” time. It will vary from the correct seconds by some value in a random interval of up to 60 seconds. In terms of random differences, the chances that all three hands ever meet will not exist.


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