Confession: This puzzle was prompted by my watch. I looked at it a short while ago and it showed precisely 6 pm. This is one of those old-fashioned watches, with two hands. So at precisely 6 pm, the two hands point in opposite directions, forming a perfect straight line.
Now this is a rare, but certainly not unique, occurrence. Thus the questions:
1) When is the next moment that the hands point in opposite directions like this, forming a straight line?
2) Between 6 pm and 6 am, but not including either, how many such straight-line moments will there be?
(Note: I don’t mean the shorter straight line formed when the hands coincide, like at noon).
Hint: One way to answer this is to consider the different angular speeds at which the hands move. The hour hand does one full rotation, 360 degrees, in 12 hours, or 30 degrees / hour. The minute hand does 360 degrees in one hour, or 360 degrees / hour. So we know how fast the angle between them is changing.
A little thought will tell you that the next straight line is at some point past 7 pm. What angle do the hands form at 7 pm? That has to change to 180 degrees.
Go for it!
Scroll down for the answers.
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Answers:
1) The next straight line will happen at 7:05:27.27 pm — ie, 5 minutes and slightly more than 27 seconds past 7pm.
2) Ten straight lines. (Not 11. Think about it.)
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