### Brain Teasers

# Line up Time

At what time after 4pm are the minute and hour and second hand of a standard clock perfectly aligned on top of each other?

### Answer

4:21 and 21 secondsHide Answer Show Answer

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## Comments

What about midnight, I mean it is definatly after 4 p.m. and all of the hands are perfectly alligned. After the high standard I've seen you expect from others I'm just currious as to why you don't view your own work with such a scrupulous eye. It's pretty hard thinking of every minute little detail to answer the most obtuse of critics isn't it MAD'ADE. People who live in glass houses ought not throw stones.

sorry the Question should imply that it meant what time directly after 4 O'clock, otherwise there would be a possibility of 11 different answers,8 of them before 12 O'clock, sorry about that

I have a question. i've never understood that quote about glass houses. Could you please explain it? maybe also explain that one about a rolling stone gathers no moss? thanks.

Do you mean people in glass houses shouldn't throw stones?

Pifle people in glass houses should not throw stones because stones break glass if you can't work that out then I really do not know how you survive on this site.

boy it's chilly in here

The answer given is incorrect. The hour hand and the minute hand line up at exactly 4:21 and 9/11 of a minute. Now 9/11 of a minute is way more than 21 seconds. It did say line up exactly! I do not know if the teaser is even possible. Certainly not between 4 and 5 O'clock.

It looks like the only correct answer is 120. At any other time, the second hand will be at some other spot on the clock when the minute hand passes over the hour hand.

jimbo's right - 4:21:21 is wrong. It should be ABOUT 4:21:49.

The formula 12/11x, where x=the hour after which the three hands of the clock align gives the answer in hours for each of the 11 occurrences for which this is true.

I think no one would argue that 1200 is one of the 11 occurrences. So to get an idea quickly that this formula is correct, think about x=11, or in other words, what is the first time that this would happen after 11 o'clock?

So using this formula with x=11: 12/11 X 11 = 12. So 12 o'clock is the first occurrence for which this is true after 11 o'clock.

Now for this teaser, x=4 (4 o'clock), so using the formula: 12/11 X 4 = 4.363636...(repeating). In other words, 4.363636...(repeating) hours or "o'clock".

But no one states the time in this way. So, to convert the decimal portion of the hours to minutes, we multiply it by 60, which gives us 21.818181...(repeating) minutes. Therefore, so far we have 4:21.818181...(repeating) o'clock as the time.

But again, no one states the time using a decimal number. So, to convert the decimal portion of the minutes to seconds, once again we multiply it by 60, which gives us 49.090909...(repeating) seconds. Therefore, rounding to the nearest second, altogether we have the time 4:21:49.

I think no one would argue that 1200 is one of the 11 occurrences. So to get an idea quickly that this formula is correct, think about x=11, or in other words, what is the first time that this would happen after 11 o'clock?

So using this formula with x=11: 12/11 X 11 = 12. So 12 o'clock is the first occurrence for which this is true after 11 o'clock.

Now for this teaser, x=4 (4 o'clock), so using the formula: 12/11 X 4 = 4.363636...(repeating). In other words, 4.363636...(repeating) hours or "o'clock".

But no one states the time in this way. So, to convert the decimal portion of the hours to minutes, we multiply it by 60, which gives us 21.818181...(repeating) minutes. Therefore, so far we have 4:21.818181...(repeating) o'clock as the time.

But again, no one states the time using a decimal number. So, to convert the decimal portion of the minutes to seconds, once again we multiply it by 60, which gives us 49.090909...(repeating) seconds. Therefore, rounding to the nearest second, altogether we have the time 4:21:49.

Time for a little thought experiment. Visualize an analog clock. When the hour, minute and second hands line up for the first time after 4 pm, all three of them will be between the numerals 4 and 5 on the clock face. In other words between 4 and 5 pm, between 20 and 25 minutes after the hour, and between 20 and 25 seconds after the even minute. If the second hand is at 49 seconds it will be way over on the left side of the face nowhere near the other two hands. Not so?

ThinksForFun has the math correct, but as geneHAG points out, the second hand is nowhere near the hour and minute hand at 4:21:49. This gives the time when the hour and minute hand perfectly line up.

The best answer for the teaser is 4:21:21, when all the hands overlap, although not "perfectly".

The best answer for the teaser is 4:21:21, when all the hands overlap, although not "perfectly".

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