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Topic: Coincident Hands (variant) (Read 514 times) |
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ThudnBlunder
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Coincident Hands (variant)
« on: Apr 16th, 2003, 12:33pm » |
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The time on my watch is exactly noon.
That is, the hour and minute hands are coincident.
What is the next time that this will occur if we consider:
(i) only the hour and minute hands?
(ii) the hour, minute, and second hands?
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| « Last Edit: Apr 16th, 2003, 1:38pm by ThudnBlunder » |
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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ThudnBlunder
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Re: Coincident Hands (variant)
« Reply #2 on: Apr 17th, 2003, 9:48am » |
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Quote:the first one is allready on the site somewhere..
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Be that as it may, (ii) is a separate puzzle, to which I have an elegant solution. 
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Chronos
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Re: Coincident Hands (variant)
« Reply #3 on: Apr 17th, 2003, 4:25pm » |
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I would imagine that the answer to (ii) is 12:00. There are only discrete times when the hour and minute hands will match exactly, and it'd be awfully weird if the second hand just happened to be exactly there at that time, too.
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Icarus
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Re: Coincident Hands (variant)
« Reply #4 on: Apr 17th, 2003, 6:09pm » |
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Indeed it would be weird. To be more precise:
Suppose in passing from one "triple crossing" to the next, the hour hand made a total of x revolutions (x possibly not an integer). The minute hand moves 12 times as fast, so it traveled 12x, and the second hand traveled 720x. Since all three are in the same spot, the difference in angle traveled between any two must be an integral number of revolutions. In particular
12x - x = 11x = k
720x- x = 719x = m
for integers k and m. So k/11 = m/719 = x. or 11m = 719k. Since both 11 and 719 are prime, 11 divides k, and 719 divides m, so x is an integer.
All three hands have made an integer number of revolutions, which puts them back at 12:00.
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| « Last Edit: Apr 17th, 2003, 6:10pm by Icarus » |
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ThudnBlunder
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Re: Coincident Hands (variant)
« Reply #5 on: Apr 18th, 2003, 4:25am » |
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Icarus, you stole my blunder, so to speak.
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