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MMM #29: Mother of all clock angle problems

Quan and Daniel have posted the answer to MMM #28 at Blinkdagger so now it's time for MMM #29.

I call this one the "mother of all clock angle problems." Some of you may have run into those problems where you have to figure out the angle between the two hands of a clock at weird times. Well, I took that problem as a starting point and added a twist to it.

Consider a 12-hour analog clock with two hands and a round face. Consider the angle between the two hands at any given time and, when the angle between the hands is not 180 degrees, take the smaller of the two angles. Thus, at 12:00 the angle between the two hands is 0 degrees. At 3:00 and at 9:00 it's 90 degrees.

If we me measure the angle between the two hands at each of the 61 consecutive minutes between 12:00 and 1:00 inclusively, what is the sum of those 61 angles?

Remember to show your work.

Here are the rules for the contest:

1. Email your answers with solutions to mondaymathmadness at gmail dot com.
2. Only one entry per person.
3. Each person may only win one prize per 12 month period. But, do submit your solutions even if you are not eligible.
4. Your answer must be explained. You must show your work! Wild About Math! and Blinkdagger will be the final judges on whether an answer was properly explained or not.
5. The deadline to submit answers is Tuesday, April 7, 12:01AM, Pacific Time. (That’s Tuesday morning, not Tuesday night.) Do a Google search for “time California” to know what the current Pacific Time is.)
6. The winner will be chosen randomly from all timely well-explained and correct submissions, using a random number generator.
7. The winner will be announced Friday, April 10, 2009.
8. The winner (or winners) will receive a Rubik’s Revolution or a $10 gift certificate to Amazon.com or $10 USD via PayPal. For those of you who don’t want a prize I’ll donate $10 to your favorite charity.
9. Comments for this post should only be used to clarify the problem. Please do not discuss ANY potential solutions.
10. I may post names and website/blog links for people submitting timely correct well-explained solutions. I’m more likely to post your name if your solution is unique.

Comments (8) Trackbacks (2)
  1. Interesting problem!

    A variation may be to measure the angles clockwise using the hour hand as reference, and constraining the angles between -180 and 180 degrees. In this case, the answer seems more interesting, and can possibly be arrived at through some intuition.


  2. How do you sum angles? For example, is 180 degrees + 180 degrees 360 degrees or 0 degrees? Is thrice 180 degrees 540 degrees or 180 degrees?

  3. Clueless – nice variation!

    Ted — to sum angles you don’t do modular arithmetic. So, 180 degrees + 180 degrees = 360 degrees. Thrice 180 degrees = 540 degrees.

  4. In this problem are we assuming the hour hand stays stationary as the time increases from 12 to 1

  5. Hi Ishita,

    No, the hour hand does not stay stationery. Just like in a real clock, the hour hand moves a little bit every minute.

  6. are we supposed to keep the direction of measurement same or reverse it after 180 degrees as the angle between the hour hand is considered to be reflex after 180 degrees?

  7. Hi Neha, I don’t quite follow your question.

    If the (acute) angle between the two hands is 90 degrees, it could also be measured as 270 degrees (the obtuse angle), right? Always pick the smaller measurement, 90 in this case.

    Does that answer your question?

  8. Firstly there are only 60 minutes from 12:00 to 1:00 and their sum is 5793


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