### Brain Teasers

# The Matching Hands of Time

During a normal 12 hour cycle (midnight to noon, or vice versa), how many times will the hour and minute hands line up exactly, and at what times, to the nearest second, will this happen? (Count midnight and noon as once only, as on a clock, they are the same time.)

### Hint

In one hour, the minute hand travels around a full circle of three hundred and sixty degrees, while the hour hand travels only thirty degrees in the same time.Also keep in mind that the minute (or hour) hand will be in the same relative position after a 360 degree rotation.

### Answer

There are 11 times during a 12 hour cycle when this happens.Here is the explanation of what times this happens:

If we let T represent the number of hours that elapse, then we can say that the minute hand moves 360 degrees per hour and the hour hand moves 30 degrees per hour. Let M be the relative position of the minute hand, and H the relative position of the hour hand, measured in degrees (relative to the 12 o'clock position).

As equations, we will have

H = 30*T, M = 360*T

Now, we can measure the angle between the two hands as M - H.

M - H = 330*T.

The hands will line up when there is no angular difference between the two. That is, either the angle is zero, or some exact multiple of 360 degrees.

So, 330*T = 360*K, where K is some whole integer value.

Hence, T = (12/11)*K, where T represents hours after 12:00.

Here, K can be any integer value from 0 to 10. If K = 0, then the time is 12:00 (the starting time). For K = 11, T = 12, which again is 12:00. The other values in between will give exact values of T in hours, which can be converted to hours, minutes and seconds. The exact times when the hands line up are:

12:00:00, 1:05:27, 2:10:55, 3:16:22, 4:21:49. 5:27:16, 6:32:44, 7:38:11, 8:43:38, 9:49:05, and 10:54:33. The next time after this will again be 12:00:00.

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## Comments

What about times like 12:33, 1:38... aren't the hands lined up at those times? I don't feel like doing the math to get the exact times, but I count 23.

I thought the idea was pretty self explanatory. By "line up", I meant that the hands point to the same spot on the clock, but if you want to work out the details for when the hands are diametrically opposite, then set the equation to 180*K instead of 360*K.

Of course, one could argue, from a strict geometric point of view, that the hands of a clock represent rays (as opposed to just line segments), so there is an implied direction. We could also argue of the semantic differences between the definitions of "lining up" and "being in a line", but I digress.

Kudos for the observation, all the same.

Of course, one could argue, from a strict geometric point of view, that the hands of a clock represent rays (as opposed to just line segments), so there is an implied direction. We could also argue of the semantic differences between the definitions of "lining up" and "being in a line", but I digress.

Kudos for the observation, all the same.

The preceding two comments demonstrate that the teaser's wording, to correspond to its intended meaning, ought to be changed to something like:

From midnight to 11:59 A.M., how many times will the hour and minute hands of a standard 12-hour analog clock point to the same spot? At what times, to the nearest second, will this happen?

I would suggest changing the second question to:

At what precise times (to the exact fraction of a minute), will this happen?

as this does make the teaser more precise and not any more difficult.

Even better, in my opinion, would be to change the teaser to require the solver to take into account that a straight line extends both ways:

From midnight to 11:59 A.M., how many times will the hour and minute hands of a standard 12-hour analog clock be in a straight line? At what precise time interval (to the exact fraction of a minute), will this happen?

The answer to the first question is NOT 23, but 22, as the answer to this 2nd question is 32+8/11 minutes (which, multiplied by 22, is 720 minutes = 12 hours)

From midnight to 11:59 A.M., how many times will the hour and minute hands of a standard 12-hour analog clock point to the same spot? At what times, to the nearest second, will this happen?

I would suggest changing the second question to:

At what precise times (to the exact fraction of a minute), will this happen?

as this does make the teaser more precise and not any more difficult.

Even better, in my opinion, would be to change the teaser to require the solver to take into account that a straight line extends both ways:

From midnight to 11:59 A.M., how many times will the hour and minute hands of a standard 12-hour analog clock be in a straight line? At what precise time interval (to the exact fraction of a minute), will this happen?

The answer to the first question is NOT 23, but 22, as the answer to this 2nd question is 32+8/11 minutes (which, multiplied by 22, is 720 minutes = 12 hours)

I thought the explanation was a bit long. Since they "line up" 11 times, and are lined up at midnight, then the minutes and seconds are each 1/11 of a hour. 60/11 = 5 4/11 = 5:27 3/11 seconds. So the times a for H=1..11: HH*5:27 3/11). No need talk about angles and degrees.

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