You are given a rectangular watch, as provided in the picture below.

enter image description here

How many times during a period of 12 hours, starting at 12:01 AM and ending at 12:01 PM, do the hour and minute hands divide the rectangle into two shapes of equal area?

  • $\begingroup$ Are the two hands pointing in opposite directions at the beginning of the 12 hours? If so, they will also be in that position at the end of the 12 hours, which means you can get an answer that differs by one depending on where the hands are when the 12 hours start. $\endgroup$ – GentlePurpleRain Aug 22 '15 at 2:49
  • $\begingroup$ Starting at 12:01 AM and ending at 12:01 PM $\endgroup$ – Moti Aug 22 '15 at 3:46
  • $\begingroup$ @Anachor Why the edit? For 2 points? $\endgroup$ – Moti Aug 22 '15 at 3:47
  • $\begingroup$ A) IMO, it looks better, B) "A clock with a rectangle" can be ambiguous, "A rectangular" watch is not C) I'm a 1k+ member, which means I no longer get 2 points for editing. If you think my edit didn't make the post better, feel free to roll back. $\endgroup$ – Rohcana Aug 22 '15 at 4:08
  • $\begingroup$ My point is that I think that with the picture it is clear what is required. I agree that for writing a book of puzzles it would be better describing it with the details you added. I think it made it (a little) better. Thanks. $\endgroup$ – Moti Aug 22 '15 at 4:14

There are:

11 times


The only times the rectangle will be split into parts with equal areas is when the hands are opposite each other, which happens 11 times during a 12 hour period (about every 1:05).

  • $\begingroup$ You are making a small logic mistake. Suggest you rethink your answer. $\endgroup$ – Moti Aug 22 '15 at 2:37
  • $\begingroup$ I simulated it with GeoGabra and calculated it separately for verification of the right answer. 12 is not the answer. $\endgroup$ – Moti Aug 22 '15 at 2:39
  • $\begingroup$ I almost had this as my original answer, but then I talked myself into 12. Need some sleep before I will be able to fully understand the math behind this. $\endgroup$ – MisterEman22 Aug 22 '15 at 3:01
  • $\begingroup$ It's not hard. assume that it happens at least once. 12 hours later, the hands are in the same position. It happens at least 11 times inbetween. Therefore the total is 12. unless I'm missing something obvious, each hour has a point opposite it on the clock face where the minute hand will be. $\endgroup$ – Kingrames Aug 23 '15 at 6:45
  • $\begingroup$ "about every 1:05" is incorrect because the time between each one changes over time. check out the difference between 5:55 and 6:00. Oh wait, I just realized I have been doing it wrong and the hour hand probably doesn't stick like it does in some clocks. Nevermind. $\endgroup$ – Kingrames Aug 23 '15 at 7:00

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