Question:
When will the hands on the clock line up again?
southpaw
2013-10-22 07:41:21 UTC
Somewhat of a classic math problem, but I cannot figure out how to solve it.

Scenario: Standard face clock with hour, minute and second hand. All three hands are aligned at 12:00:00.

Q: What will the next time be that all three hands are (exactly) aligned again? I know the answer, but can't figure out how to get there. All I can think of is one must calculate when the three hands have had the same amount of travel in degrees or radians?

The answer (so I have read) is they will all re align at 5:27:27, tho your calculations may dispute that. Thanks.
Five answers:
2013-10-22 09:30:31 UTC
the minute hand has velocity

a = (2 pi) / 3600 radians per second

hour hand

b = (2pi) / 43200 raidans per second



hour hand starts effectively 2pi ahead of minute hand, so time for minute hand to catch is



at = 2pi + bt



t = 2pi / (a - b) = 2pi / [(2 pi) / 3600 - (2pi) / 43200 ] = 2pi / [22 pi / 43200]

= 43200 / 11 seconds



so hour and minute coincide every 43200 / 11 seconds



where the hour hand angle is b x n x 43200 / 11 = ( (2pi) / 43200) x n x 43200 / 11



= 2 pi n / 11



then we need second hand position equal



second hand velocity = 2 pi / 60



so we need



n(43200 / 11) (2 pi / 60) mod 2pi = 2 pi n / 11
DWRead
2013-10-22 15:09:29 UTC
You can tell right away that 5:27:27 is incorrect. The minute hand will not be aligned with the second hand, because it will be almost halfway between 27 minutes and 28 minutes. The number of minutes can't be the same as the number of seconds, unless both are 0. That's why Scott's answer can't be correct.



After t seconds, the second, minute, and hour hands have traveled 6t, t/10, and t/120 degrees, respectively.

The angles repeat every 360°, so you want t such that 6t ≣ t/10 ≡ t/120 (mod 360).
2013-10-22 15:32:01 UTC
The hands will line up again at 1:05:05 or five minutes and five seconds after one o'clock. In fact the hands will line up every hour: At 2:10:10, at 3:15:15, 4:20:20, etc, etc.
Bullwinkle
2013-10-22 14:54:14 UTC
Here are two "discussions" of the problem:



http://mathforum.org/library/drmath/view/56819.html



and



http://puzzles.nigelcoldwell.co.uk/thirtyfive.htm



euclid
?
2013-10-22 14:48:37 UTC
1:05:05


This content was originally posted on Y! Answers, a Q&A website that shut down in 2021.
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