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Old 2011-09-22, 19:06   #1
Flatlander
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Default Watch Hands.

Is it possible for all three hands of a watch to be exactly 120 degrees apart?
If so, at what time?
(Plagiarized from BBC Focus magazine.)
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Old 2011-09-22, 19:38   #2
firejuggler
 
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does your clock count in 24 hours or in 12 hours cycle
(joke answer : no, since all clock are now digital)?

12 hours case :
let's see.. day has 24 hours, divided into 2 cycle of 12 hours. meaning that each hours, the hour hand move by 360/12 : 30 degree.
each hours is composed by 60 minutes , meaning a move of 6 degree by minute, moving the hour hand by 30/60, 0.5 degree
each minute is composed of 60 second, moving the minute hand by 0.1 degree and the hour hand by 0.008333 degree
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Old 2011-09-22, 19:46   #3
Flatlander
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It's just a standard analogue watch with numbers 1 to 12, hour, minute and second hands.
It's not a trick question.
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Old 2011-09-22, 20:12   #4
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Quote:
Originally Posted by Flatlander View Post
It's just a standard analogue watch with numbers 1 to 12, hour, minute and second hands.
It's not a trick question.
What is the granularity with which the clock hands move? (A "perfect" analog clock has infinitesimal granularity. Real clock hands "jump" with varying granularity for different hands)

In the perfect case, the answer is no. But it gets real close.
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Old 2011-09-22, 20:27   #5
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another precision then, does the minute hand move each second (by increment of 0.1degree) or only on the 60th second?
if it is the later, then 4H00 and 40 seconds is an answer
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Old 2011-09-22, 21:24   #6
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I think we should assume all the hands jump 'instantaneously' every second.

The magazine gives the best solution.

Last fiddled with by Flatlander on 2011-09-22 at 21:24
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Old 2011-09-23, 04:52   #7
LaurV
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if the clock moves with a granulation of 360 milliseconds (all hands instantly jump at these intervals), assume the clock starts at 0:00:00, then the best "alignment" is after 32725440 milliseconds, you can compute what time is then. This alignment does not improve if lower time step is taken, as 36, 18, 10, 5, 1 millisecond.
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Old 2011-09-23, 11:00   #8
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Quote:
Originally Posted by LaurV View Post
if the clock moves with a granulation of 360 milliseconds (all hands instantly jump at these intervals), assume the clock starts at 0:00:00, then the best "alignment" is after 32725440 milliseconds, you can compute what time is then. This alignment does not improve if lower time step is taken, as 36, 18, 10, 5, 1 millisecond.
That's just after 9 o'clock. The solution in the magazine is nowhere near that.
Maybe you solved something else?

edit:
As my maths teacher would say 'full points for showing your working-out'.

Last fiddled with by Flatlander on 2011-09-23 at 11:06
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Old 2011-09-23, 11:55   #9
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Default OK Miss!

Quote:
Originally Posted by Flatlander View Post
As my maths teacher would say 'full points for showing your working-out'.
At m minutes past h, the little hand has rotated 30(h + m/60) degrees.
The big hand has rotated 6m degrees.

We want to find 24 values of m such that 6m - 30(h + m/60) = 120 or 240
for twelve integers h.

The second hand has gone through 360(m - INT(m)) degrees.

Finding the best of the 24 cases is left as an exercise for the teacher

David

Last fiddled with by davieddy on 2011-09-23 at 12:30
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Old 2011-09-23, 12:49   #10
LaurV
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You won't believe, but I produced that value in less then 4 minutes, using Windows's pocket calculator. And I was amazingly right!

That is because there is a simple trick to get the angle between the vertical axis and (any of the) hands of the clock if you specify the time in seconds. I met an equivalent problem in the past. Here I used 360 milliseconds trying to be more "accurate", and to get rid of the divisions with the angles, because the 360 degrees in a circle and 3600 seconds in a hour, and that was a mistake, because the REAL time highly depends of the accuracy of the movement of your clock. When your clock is moving continuously, the most perfect match you get is INDEED at

09:05:25.438308786993808 (milliseconds: 32725438.308786993808)

with an accuracy of a femtosecond.

When I arrived home from work (19:45 here), I checked everything with pari/gp. It took me longer to put the program in a "postable" form (initial version was just a "obfuscated" 3-lines function, directly typed in), add comments and make tests. I had to introduce a printf() to output it in nice format, otherwise some guys here would jump on my head relating to my programming style. Some functions are now quite slow, but more clear. I also had to add the "start", "stop" and "flags" parameters to be able to "refine" the search to femtosecond, otherwise to run it for all the 12 hours would take ages!!

Chech for yourself.

(mark all, save as clock.gp, run it in pari)

edit: at the end of the code there are the results of the testing, so if you don't want to run it, scroll down and read

Code:
/*the output is padded for 120 characters screen width*/
clock(step_ms,start,stop,flags)= /*moving step in milliseconds, real*/
{
    if(step_ms>3600000, print("Use steps smaller then 1 hour, please!"); return(0));
    /*if(step_ms<0.001, print("Use steps larger then 1 microsecond, please!"); return(0));*/

    step=start+0.0;
    epsilon=3.*360^2;
    while(step<min(12.*60*60*1000,stop),
        /*angle between hours hand and vertical*/
        hoursangle=360.*step/(12*60*60*1000);
        
        /*angle between minutes hand and vertical*/
        minutesangle=realmod(360.*step/(1*60*60*1000),360);
        
        /*angle between seconds hand and vertical*/
        secondsangle=realmod(360.*step/(1*1*60*1000),360);
        
        if((a= ((ahm=computeangle(hoursangle,minutesangle))-120)^2+
               ((ahs=computeangle(hoursangle,secondsangle))-120)^2+
               ((ams=computeangle(minutesangle,secondsangle))-120)^2)<=epsilon,
            savetime=step;
            epsilon=a;
            saveahm=ahm;
            saveahs=ahs;
            saveams=ams;
            if(bitand(flags,1),print([savetime,a]));
            if(bitand(flags,2),
                printf(" Time: %02d:%02d:%02d.%06d,  H-M angle: %8.4f,  H-S angle: %8.4f,  M_S angle: %8.4f,  Delta: %1.9f\n",
                    (tt=hmsms(savetime))[1],tt[2],tt[3],tt[4]*1000,ahm,ahs,ams,epsilon)
            )
        );
        step+=step_ms
    );
    printf("\n Time: %02d:%02d:%02d.%06d,  H-M angle: %8.4f,  H-S angle: %8.4f,  M_S angle: %8.4f,  Delta: %1.9f",
        (tt=hmsms(savetime))[1],tt[2],tt[3],tt[4]*1000,saveahm,saveahs,saveams,epsilon);
    return([savetime,epsilon]);
}

/* Mod() function for reals, did not find some analog periodic function in pari/gp */
realmod(x,y)=
{
    while(x>=y,x-=y);
    return(x);
}

/*smallest angle between handles, circular to 360*/
computeangle(x,y)=
{
    my(a);
    return(if((a=abs(x-y))<=180, a, 360-a));     
}

/*convert a (real) number of milliseconds to hh_mm_ss_ms format*/
/* (ugly, but more clear)*/
hmsms(n)=
{
    my(v);
    v=vector(4);
    v[1]=truncate(n\3600000);
    v[2]=truncate((n-v[1]*3600000)\60000);
    v[3]=truncate((n-v[1]*3600000-v[2]*60000)\1000);
    v[4]=n-v[1]*3600000-v[2]*60000-v[3]*1000;
    return(v);
}

/*
results:

1. best result, last refined (femtoseconds) with cmd line:
  
  > clock(0.000000000001,(9*3600+5*60+25.438308786993)*1000,(9*3600+5*60+25.438308786994)*1000,1)
    Time: 09:05:25.438308786993808 (milliseconds: 32725438.308786993808000)
        H-M angle   : 119.8318449721410990666666667
        H-S angle   : 120.0821331845954337333333332
        M_S angle   : 120.0860218432634672000000001
        Delta       : 0.04242173092445985941909721982

2. assume the clock is moving with 360 ms resolution 
  (fits better with the 360 angle and 3600 seconds in an hour)
  (also valid for all divisors of 360, except 1,2ms, the time won't change)
  (for all other numbers lower then 360 which are not divisors, 
   like 100ms, etc, then delta is worse)
   
  > clock(360,0,12*3600000,3)
    Time: 09:05:25.440000 (milliseconds: 32725440.000000)
        H-M angle   : 119.832000000
        H-S angle   : 120.072000000
        M_S angle   : 120.095999999
        Delta       : 0.04262399999999999999999997217

3. assume the clock is moving with 1 or 2 ms resolution, there will be intermediary accuracy,
   between the two above, up to the closest "integer", but the time stays around 9:05:25. 
  
4. for 300ms, another time come quite close, with an insignificant difference of delta:
    Time  : 02:54:34.500 (milliseconds: 10474500.000000)
    Delta : 0.3115624999999999999999999879

    in this case, delta for 9:05:25.500 is still smaller:
    Delta : 0.3115624999999999999999999877

5. for 400 ms, these two times beats everything (our former times are much behind 
   with their deltas around 200 to 800... see what few milliseconds can do! :D)
    Time1  : 06:10:50.800 (milliseconds: 22250800.000000)
    Time1  : 05:49:09.200 (milliseconds: 20949200.000000)
    both have a delta of: 0.5848222222222222222222234173

6. when we assume that the clock is moving with half second resolution, as for
   blinking dots of digital clocks, we find the same case as case 4 (300ms, is LCM) 

7. 600ms brings a delta of 1.06, for time: 02:32:52.8
   all others are much behind

8. 700ms is closed to case 5, but better, 6:10:50.9, delta: 0.2083

9. 720ms (as a multiple of 360, case 2) fells exactly over it, delta 0.04xxx
   and this is valid for other higher multiplies too

10. 800ms is case 7 above (600ms)

11. The granulation of 1 second brings a new time. Seems that our old values are 
    "jumped over", by the larger granulation

  > clock(1000,0,12*3600000,3)
    Time: 05:49:09.000000 (milliseconds: 20949000.00000000000000000000)
        H-M angle   : 120.3250000000000000000000000
        H-S angle   : 120.5750000000000000000000000
        M_S angle   : 119.1000000000000000000000000
        Delta       : 1.246249999999999999999999951

  note that for 3 seconds we will get the same, this time is a multiple of 3 seconds

12. for 2 seconds we get: Time: 03:37:58 (13078000) with a bigger delta: 2.4072

13. Higher times: for 5 seconds we get: Time: 03:59:40 (14380000) with a big delta: 7.3888
    and the same for 10, 20 seconds, as this time is multiple of them. But not for 
    15 seconds, that is a 6:55:15, with a delta of 22.9063. The half minute bring us to 
    10:10:30 (delta=41.625) and the full 60 seconds shows 8:21:00 (delta=166.5). Note that
    except the last, even these slow motion clocks reach the 120 angles quite well, with 
    a variation not bigger then 5 degrees :P

and so on, you play with it, I stop here because I like number 13...
*/

Last fiddled with by LaurV on 2011-09-23 at 13:07
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Old 2011-09-23, 13:40   #11
Flatlander
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Here's the best fit the magazine gives:
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