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Superimposition (Posted on 2004-01-30) Difficulty: 3 of 5
At exactly 12:00, the hour, minute, and second hands are all pointing straight up.

Are there any other times that all three hands are superimposed?
If so, when?

See The Solution Submitted by DJ    
Rating: 3.4000 (10 votes)

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Solution solution | Comment 3 of 10 |
The hour and the minute hand are together once every 12/11 hour. The number of hours past noon is the first column on the below table; then the time in h:mm:ss.sss format and the position in degrees clockwise from the 12-o'clock position of the hour-, minute- and second-hands:

1.091 1:05:27.27 32.72727273 32.72727273 163.63636364
2.182 2:10:54.55 65.45454545 65.45454545 327.27272727
3.273 3:16:21.82 98.18181818 98.18181818 130.90909091
4.364 4:21:49.09 130.90909091 130.90909091 294.54545455
5.455 5:27:16.36 163.63636364 163.63636364 98.18181818
6.545 6:32:43.64 196.36363636 196.36363636 261.81818182
7.636 7:38:10.91 229.09090909 229.09090909 65.45454545
8.727 8:43:38.18 261.81818182 261.81818182 229.09090909
9.818 9:49: 5.45 294.54545455 294.54545455 32.72727273
10.909 10:54:32.73 327.27272727 327.27272727 196.36363636
12.000 12: 0: 0.00 0.00000000 0.00000000 0.00000000

----
In none of these instances does the angle of the second hand match the angle of the other two hands except at 12. Of course you could say that midnight is different from noon, but both are 12 o'clock.
  Posted by Charlie on 2004-01-30 09:39:00
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