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BGonline.org Forums
OT Analog clock math problems
Posted By: Tom Keith In Response To: OT Analog clock math problems (Jason Lee)
Date: Tuesday, 4 May 2010, at 5:36 p.m.
Here's my stab at #2.
Let's looks specifically at hour hand, minute hand, and second hand in that order.
Define a "long hour" as the minimum time it takes the minute and hour hands to go from pointing in the same direction to again pointing in the same direction. This happens 11 times in 12 hours, so a "long hour" is 12/11 of a real hour.
A long hour is also how long it takes for the hour and minute hands to go from being +120 degrees apart to again being +120 degrees apart. The first such event happens at 1/3 of a long hour after 12:00 (which is 4/11 of a real hour).
Now define a "long minute" as the minimum time it takes the minute and second hands to go from pointing in the same direction to again pointing in the same direction. This happens 59 times per hour, so a "long minute" is 60/59 of a real minute.
A long minute is also how long it takes for the minute and second hands to go from being +120 degrees apart to again being +120 degrees apart. The first such event happens at 1/3 of a long minute after 12:00:00 (which is 20/59 of a real minute).
Note that the hour-minute thing happens only at multiples of 1/11 of a clock revolution and the minute-second thing happens only at multiples of 1/59 of a clock revolution. Because 11 and 59 are relatively prime, the only time both events could happen together is at zero revolutions (i.e., at 12:00:00). But at 12:00:00 all the hands are pointing in the same direction so they are obviously not 120 degrees apart.
Presumably the same argument applies if you look at the hands going in the other order.
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