Shollenbarger's Clock

  1. Shollenbarger's Clock:

    Excluding 12:00, Is there ever a time on a clock where the second, minute and hour hands are equidistant from each other? (In other words, the three hands are all 120° apart? If yes, give the EXACT time(s) that this occurs. If no, prove that no such time exists. Good Luck!
     
  2. jcsd
  3. -------------------
    If you include 12:00
    -------------------

    The second hand could be on 12:00, the minute hand on 4:00 and the hour hand on 8:00

    Time would be 8:20

    --------------------
    If you exclude 12:00
    --------------------

    I personally don't think it's possible.

    If any four hour marks distance stand for 120° (each hour mark is 30°), then 12:00 must be included.

    Thus no such time exists!
     
    Last edited: Jun 17, 2006
  4. But, at 8:20:00, the hour hand of a clock doesn't point directly at the 8. In fact, it will be one third of the way from 8 to 9. Thus, at 8:20:00 the hands will not all be [tex]120^\circ[/tex] apart.

    I think he was saying to exclude the time 12:00:00 from consideration, rather than the direction of pointing at the 12.
     
  5. Parlyne is correct. I wanted to exclude 12:00:00 from consideration as an answer, because all three hands are on top of each other. But one hand can be pointing at 12.

    8:20:00 is not a valid solution:

    From second to minute: 120°
    From minute to hour: 130°
    From hour to second: 110°
     
    Last edited: Jun 17, 2006
  6. What about:

    hour hand on 4:00, minute hand on 12:00 and second hand on 8:00

    Time = 4:00
     
  7. This time is: 4:00:40. This is not a valid solution:

    From minute to hour: 119 2/3°
    From hour to second: 119 2/3°
    From second to minute: 120 2/3°

    The hour hand cannot be at 4:00 or the minute hand at 12:00 when the second hand is at 8:00, because it would now be 40 seconds after 4:00, thus the hour hand and the second would have moved slightly.

    Keep trying. (Fractions of a second are allowed as valid answers, if needed.)
     
    Last edited: Jun 17, 2006
  8. My last attempt of the day is:

    Hour hand on 3:00

    Minute hand on 0:59

    Second hand on 0:00:39

    Time = 3:59:39
     
  9. No, unfortunately 3:59:39 is not a valid answer:

    From minute to hour: 121.925°
    From hour to second: 114.175°
    From second to minute: 123.9°
     
  10. its not possible.......one way of proving it is assume the time as xhrs, yhrs, and z secs........and put appropriate conditions on them... then find angles and solve..........
     
  11. This is a very messy explanation. "Put appropriate conditions on them" doesn't give us the reason.

    What conditions?
     
  12. I too believe it's impossible. I haven't finished the proof, but the idea is that I make the hour, minutes and seconds hands of the clock into unit vectors which are a function of time, then take dot products and set them equal (because the dot product of any two of the aforementioned vectors must be |1|*|1|*cos[120 deg], i.e. 4 hour-marks apart) and finally show that there's no way that that's possible. There might be an easier way, but I like vectors.

    Now here's another brainteaser, if this one turns out to be impossible: what time on Shollenbarger's Clock will make the hands the closest to 120 degrees?
     
    Last edited: Jun 18, 2006
  13. DaveC426913

    DaveC426913 16,244
    Gold Member

    Correct me of I'm wrong, but no one said they have to be at exact integer times, right?

    Don't these three hands line up at 120 degrees from each hour approximately 11 times in every 12 hour period? (I'll leave it to others to determine the exact times.)
     
  14. No. The hour and minute hands line up 11 times in each 12 hour period. The second hand will only be in line with these one of those 11 times.
     
  15. I'm going to go out on a limb and say that there is no such time. To demonstrate, let's assume there is a time when the hands are all [tex]120^\circ[/tex] apart. Call the hour x, the minute y and the second z. Because of the format times are cited in, x and y must be integers.

    Using these, we can express the angle at which each hand sits. The hour hand sits at [tex]\alpha = \left (\frac{x}{12} + \frac{y}{720} + \frac{z}{43200} \right)*360^\circ = 30x + \frac{y}{2} + \frac{z}{120}[/tex]. The minute hand sits at [tex]\beta = \left ( \frac{y}{60} + \frac{z}{3600} \right )*360^\circ = 6y + \frac{z}{10}[/tex]. And, the second hand sits at [tex]\gamma = 6z[/tex].

    We set the second and minute hands [tex]120^\circ[/tex] apart by setting [tex]\beta = \gamma \pm 120[/tex]. This gives the condition [tex]y = \frac{59}{60} z \pm 20[/tex]. Since y must be an integer, z must be an integral multiple of 60/59. Thus, we can write [tex]z = \frac{60}{59}n[/tex] and [tex]y = n \pm 20[/tex], where n is an integer.

    Finally, to put the hour hand in the correct position, we set [tex]\alpha = \gamma \pm 240[/tex]. This reduces to the condition that [tex]x = \frac{11}{59} n \pm \frac{23}{2}[/tex].

    We can see that, for x to be an integer, n must be valued such that [tex]\frac{11}{59} n[/tex] is an odd half integer. But, since 59 is odd and 11 and 59 are relatively prime, the only way for [tex]\frac{11}{59} n[/tex] to be an odd half integer is if n is an odd half integer multiple of 59 - [tex]n = \frac{2m+1}{2} 59[/tex]. Since all factors in the numerator are odd, this can't be an integer. Thus, if n is an integer, x is not.

    Thus, there are no x, y, and z satisfying the relationships above such that x and y are both integers.
     
  16. conditions like:
    x,y,z should be integers...........or atleast x and y should be integers(if u want to accept decimal seconds). further 0<= x <12....... 0<= y <60 and 0 <= z < 60........with these conditions u'll never get an appropriate answer
     
  17. It gets close enough for practical purposes at

    02:54:34.576
     
  18. I tried to clarify this earlier...

    Decimal seconds are allowed. The seconds do not have to be integers. But give your answers with fractions.

    Unfortunately, this is a brain teaser and not a practical problem. So, 'close enough' is not equidistant.
     
  19. The answer is that the three hands never form 120 degree angles w.r.t each other. I have an inelegant solution. There are in fact, 22 times in a 12 hour period that the hour and minute hands are 120 degrees apart. It suffices to determine where the second hand is at those times. I did so and found that the second hand is never at the right place at the right time. Here are the results of a computer run. The first column is the time. The second column is the angle between the hour and minute hands. The third column is the angle between the minute and second hands. Take the first pair of lines for instance. Sometime between 00:21:49 and 00:21:50, the hour and minute hands were 120 degrees apart. However, at no time during that interval was the second hand near to 120 degrees from the minute hand. There is some symmetry in the data so that it turns out that with a little more analysis, you wouldn't need to check all 22 times. The closest approach is at 2:54:32 (and at 3 other times).

    00:21:49 -119.9917 -163.1000
    00:21:50 -120.0833 -169.0000

    00:43:38 -239.9833 33.8000
    00:43:39 -240.0750 27.9000

    01:27:16 -119.9667 67.6000
    01:27:17 -120.0583 61.7000

    01:49:05 -239.9583 264.5000
    01:49:06 -240.0500 258.6000

    02:32:43 -119.9417 -61.7000
    02:32:44 -120.0333 -67.6000

    02:54:32 -239.9333 135.2000
    02:54:33 -240.0250 129.3000

    03:38:10 -119.9167 169.0000
    03:38:11 -120.0083 163.1000

    03:59:59 -239.9083 5.9000
    04:00:00 120.0000 0.0000
    04:00:01 119.9083 -5.9000

    04:43:38 -119.9833 33.8000
    04:43:39 -120.0750 27.9000

    05:05:27 120.0250 -129.3000
    05:05:28 119.9333 -135.2000

    05:49:05 -119.9583 264.5000
    05:49:06 -120.0500 258.6000

    06:10:54 120.0500 -258.6000
    06:10:55 119.9583 -264.5000

    06:54:32 -119.9333 135.2000
    06:54:33 -120.0250 129.3000

    07:16:21 120.0750 -27.9000
    07:16:22 119.9833 -33.8000

    07:59:59 -119.9083 5.9000
    08:00:00 240.0000 0.0000
    08:00:01 239.9083 -5.9000

    08:21:49 120.0083 -163.1000
    08:21:50 119.9167 -169.0000

    09:05:27 240.0250 -129.3000
    09:05:28 239.9333 -135.2000

    09:27:16 120.0333 67.6000
    09:27:17 119.9417 61.7000

    10:10:54 240.0500 -258.6000
    10:10:55 239.9583 -264.5000

    10:32:43 120.0583 -61.7000
    10:32:44 119.9667 -67.6000

    11:16:21 240.0750 -27.9000
    11:16:22 239.9833 -33.8000

    11:38:10 120.0833 169.0000
    11:38:11 119.9917 163.1000
     
  20. Not quite. The second hand gets closer to the 120 degree position a couple of seconds after this time (and as the second hand goes round sixty times faster than the minute hand, the overall accuracy is better even though the minute hand is then a little off). That was the basis of my post, but apparently, it's not the answer the OP was looking for.
     
  21. You misinterpretted what I meant. Of the 22 times that the hour and minute hands are exactly 120 degrees apart, 2:54:32 finds the second hand at the closest to 120 degrees from the minute hand. Of course, there are other times that the second hand is exactly 120 degrees from the minute hand, but none of them are among the 22 times in my small universe.
     
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