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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?

  Submitted by DJ    
Rating: 3.4000 (10 votes)
Solution: (Hide)

Well, the first time the hour and minute hands are superimposed is "12/11" o'clock (shortly after 1:05), or 1/11 hour after 1:00. Where is the second hand then? At 1:00 the second hand was on 12. In 1/11 hour, it goes 3600/11 seconds or 327+3/11 seconds, or 5 times around the clock + 45+3/11 seconds. So at 12/11 o'clock, the second hand is 3/11 seconds above the 9 digit. The next time that the hour and minute hands coincide are at 24/11 (shortly after 2:10). And the second hand is 6/11 seconds past the 6 digit. If you check all 11 times when the minute and hour hands coincide, you will see that all three hands never coincide.

You can also reason this out on the basis of symmetry. If all three hands coincide at some time n/11, then they must also coincide at some time (11-n)/11. If there are only two times where they coincide (including 12:00), then these two times must be 180 degrees apart on the clock. There is no time of the form n/11 that is half of the way past 12:00. Similarly, the three hands cannot coincide three times, or four times, or five times, etc. All of these numbers are relatively prime to 11. Only 11 times could possibly work, and we eliminated that above. So 12:00 is the only time when the three hands coincide.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionK Sengupta2007-08-29 12:10:23
Some Thoughtsmaybe...logischer Verstand2004-05-03 22:35:48
of courseBrian2004-03-04 22:03:06
re(3): solutionSilverKnight2004-03-02 12:32:08
re(2): solutionhunter2004-03-02 07:10:09
Solutionre: solutionSilverKnight2004-01-30 13:45:41
I Say NoRichard2004-01-30 12:35:46
SolutionsolutionCharlie2004-01-30 09:39:00
SolutionIntegersJils2004-01-30 09:27:50
EquationsJils2004-01-30 08:45:09
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