### 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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