Logical thinking questions-hour/minute/second

# Does the hour-hour/minute-hour timing overlap as if it were a high school physics question? #

 

 

Example:

24 hours a day. How many times does the hour and minute hands of the clock overlap?

Answer: 22. Within 24 hours, the needle turns around 24 times, and the hour hand turns around 2 times (which is easy to ignore and leads to a coincidence of 24 times), which is similar to the pursuit problem. Let's take a closer look.

Assume that the clock starts from. At this time, the hour hand and the minute hand are overlapped, so there must be a coincidence every hour before. When is approaching, the hour hand has almost completed a round of journey to reach 12 points, so they happen to overlap at 12 points, over 12 points, the hour hand began a new circle.

So the coincidence time is: 12 points, more than 1 point, more than 2 points, more than 3 points, more than 4 points, more than 5 points, more than 6 points, more than 7 points, more than 8 points, more than 9 points, more than 10 points. 11 times in total.

The new circle is 11 times, so it is 22 times.

　　

Based on the above questions, we will increase the difficulty of the questions.

There are 24 hours a day. How many times are the hourly, minute, and second hands of the clock overlapped during these 24 hours?

　　　　

Answer: This question requires not only the hour hand and the minute hand overlap, but also the second hand to join in the fun, to overlap together, is 3 p.

We have discussed the hour and minute hands before, because we can overlap the three needles on this basis.

Set the hourly speed to W, so the sub-needle angular speed is 12 W, and the second needle is 720 W.

Set the hour and minute hands to overlap at angle X. the time consumed by the hour hands is X/W, and the time consumed by the minute hands is (x + 2npi)/(12 W ), n is the number of times the minute hand exceeded the hour hand. It must be an integer multiple of 360 degrees and the time consumed is the same.

So: X/W = (x + 2npi)/(12 W), x = (2npi)/11; From the above we can see that N values are 1, 2, 4 ,...... 22

Similarly, the relationship between the hour hand and the second hand is: X/W = (x + 2mpi)/(720 W), x = (2mpi)/719; m is the second hand that exceeds the hour hand's circle.

The above two formulas can obtain (2npi)/11 = (2mpi)/719, M = (719/11) n, because m must be an integer, therefore, n can only be a multiple of 11, 11 and 22.

Therefore, the three-pin coincidence is only twice, with 0 points and 12 points respectively.

　　

 

　　

　　

Logical thinking questions-hour/minute/second