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Topic: Hexaclock (Read 382 times) |
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Noke Lieu
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pen... paper... let's go! (and bit of plastic)
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You may have seen them. A hexagon-based-pyrami-clock, with sections of them to denote hours, minutes, seconds. They all rotate around the centre of the pyramid at the appropriate speed.
How often does it reform into a smooth pyramid?
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a shade of wit and the art of farce.
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Three Hands
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Re: Hexaclock
« Reply #1 on: Apr 30th, 2004, 9:56am » |
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Well, it certainly reforms every 2 hours, but I suspect it may reform more frequently than that. Might have to think a bit more about this one...
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Grimbal
wu::riddles Moderator
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Re: Hexaclock
« Reply #2 on: May 19th, 2004, 6:40am » |
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This kind of clock is typical of a manager's desk. I never suspected what kind of difficult problems they have to solve daily.
OK. Here is my proposal:
In 2 hours, the hour segment turns 1/6 turn and the minute segment turns 2 turns, or 12/6 turns. The speed of the minutes relative to the hours is 11/6 turns in 2 hours. So, in 2 hours, the two segments match 11 times.
In the same time, the seconds segment turns 2*60=120 turns or 720/6 turns. Relative to the hour segment, it is 719/6 turns. The hours and seconds segments match 719 times in 2 hours.
Since 719 and 11 are relative primes, the hours never match simultaneously the minutes and the seconds segment, except at the start of a 2 hours cycle.
So, the obvious match every 2 hours is all you get.
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Noke Lieu
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pen... paper... let's go! (and bit of plastic)
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Re: Hexaclock
« Reply #3 on: May 20th, 2004, 8:45pm » |
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Now THAT'S an answer.
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a shade of wit and the art of farce.
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Three Hands
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Re: Hexaclock
« Reply #4 on: May 21st, 2004, 1:53am » |
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No wonder I couldn't think of any other times the pyramid would reform...
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