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Topic: clock puzzle...
Replies: 17   Last Post: Aug 2, 2003 7:00 PM

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Posts: 4
Registered: 12/13/04
Re: clock puzzle...
Posted: Aug 2, 2003 12:33 PM
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Hi James,

"James Waldby" >
> I think your area formula went awry -- the values I see from it are
> about 0.444, (presuming 11 Pi s / 21600 etc are in radians) rather
> than in the neighborhood of 1.2990381, the area of a triangle
> inscribed in a circle of radius 1. But that aside, although
> 05:49:09.1233840851 and 06:10:50.8766159149 aren't too bad,
> both 02:54:34.56169071* and 09:05:25.43830908 are much better,
> giving areas about .000008 bigger. (More exactly,
> area 1.2990353071, angles 119.831845018 120.082130171 120.086024811
> vs. 1.29902692089, angles 120.336310208 119.835723690 119.827966102)
> * About the same time that Robert Israel mentioned, in his
> derivation via analysis and number theory.
> The four times mentioned above are the only ones that have neighborhoods
> giving area > 1.299; from an exhaustive binary search, there are 16 time
> neighborhoods that give areas between 1.298 and 1.299, eg 08:00:20.17865
> with area 1.29870.
> -jiw

Ah well, my formula is nearly correct. I made a typo with one of the signs,
it should be

(Sin[11 Pi s / 21600] + Sin[59 Pi s / 1800] - Sin[719 Pi s / 21600] ) / 2

This I get to be 1.2989558 when s = 20949. This is not too far from 3
Sqrt[3] / 4 = 1.29903811

I then made the following mistake. A simple search using integer seconds
values gave 20949 sec as a maximum. I then assumed incorrectly that using
that value as a starting point would give the best value.

For a clock that only displays seconds, like many clocks, I think that the
values 20949 & 22251 are quite good. However for a clock that displays
seconds smoothly (for example, like mains driven ones) then the solution
afforded by Robert Israel is correct.

I once repaired a quartz clock that incremened in 1/2 seconds. I think the
best times for these are 10474.5 & 32725.5 = 02:54:34.5 & 09:05:25.5.

Sorry for all the errors.

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