2016 comeback

Algebra Level 5

This 2016, when was the 201 6 th 2016^\text{th} instance that the hour hand and the minute hand overlap?

Write your answer as mmddhhmmss in 24 hour format and rounded off to the nearest second.

For example, if your answer is July 3, 2016, 5:15:23 pm , then write your answer as 0703171523.

Note: The first overlap is on Jan 1st midnight, namely 0101000000.

The answer is 401141055.

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1 solution

Efren Medallo
Jul 3, 2016

First, in a 24 hour cycle, we need to find out how many times the minute and hour hand overlap. To find out, we have to determine the relationship between the sweeps made by the hands.

The hour hand travels at 30 deg h r \frac{30\deg}{hr} , while the minute hand travels at 360 deg h r \frac{360\deg}{hr} . After 0:00 (12:00 midnight of the day), the hour hand and the minute hand will meet again a little after 1:00.

Exactly, this is determined by

30 x = 360 x 360 ( y ) 30x = 360x - 360(y)

where y y denotes the number of full rotations the minute hand has passed, which, just right after 1:00, is 1. We subtract it because at this time x x , the minute hand will have completed y y full laps already.

In this case, x x becomes 12 11 \frac{12}{11} hours from 0:00. In the same manner, we can determine the next time the hands meet, substituting y y with 2 2 . Since we like to know how many times these overlaps occur in a day, we use x = 24 x = 24 and solve for y y , giving us y = 22 y=22 . This means after 0:00 of day 1, there are 22 overlaps, but including 0:00 of day 2. Thus, there are just a total of 22 overlaps in a day.

So for this question, we may now find at what day the 2016th overlap occurs. Dividing 2016 by 22 gives 91 with a remainder of 14. This means the 2016th overlap falls on the day 91 days after January 1, which is April 1.

A remainder of 14 means that it falls on the 14th occurence within the day, or, based on the equation, is the 13th occurence after 0:00 of April 1. Solving for this gives us 2:10:55 pm, or 14:10:55. Thus, the final answer is

April 1 at 14:10:55

0401141055 0401141055

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