Brain-Teaser 717: Wrong hands
From The Sunday Times, 13th April 1975 [link]
Waking in the night I glanced at my bed-side clock and thought it indicated just after 2:20. Putting on my spectacles and looking more carefully I saw that it was actually just after 4:10. I had, of course, interchanged the hands at my first glance.
I began to wonder what time around then that my observation would have occurred, to the exact fraction of a minute, if the hands could have been exactly interchanged.
What do you think?
This puzzle is included in the book The Sunday Times Book of Brain-Teasers: Book 1 (1980). The puzzle text above is taken from the book.
[teaser717]
Hugh Casement and 
S2T2 
Jim Randell 8:26 am on 25 March 2021 Permalink |
I we divide the circle of the clock face into 60 sectors, then the minute has travels at 1 sector/minute, and the hour hand at 1/12 sector/minute.
If we suppose the difference (in minutes) between the times is d, then the hour hand has moved a distance d / 12 sectors (which is roughly 1/6 of the circle), and the minute hand has moved distance d sectors (and roughly twice – 1/6 times around the circle):
So the difference between the times is nearly 111 minutes, about what we would expect.
If we suppose the actual time is m minutes after 4 o’clock (i.e. 240 + m minutes), then the minute hand will be showing m minutes, which is m sectors around the circle.
And at the mistaken time = (240 + m − d) the hour hand would be showing the same point:
Solution: The exact time would be 11 + 107/143 minutes after 4 o’clock.
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Hugh Casement 1:45 pm on 25 March 2021 Permalink |
I make that 11 minutes, 44 and 128/143 seconds,
mistaken for 20 minutes, 58 and 106/143 seconds (20 and 140/143 minutes) after 2.
There’s a lot to be said for digital clocks, especially in the middle of the night!
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