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Sunday, July 27, 2008

Confusing Clock

The story goes that when the Swiss didn't have so much experience yet with making clocks, a painful mistake was made with a church clock. The clock was officially put into use when it showed 6 o'clock. But soon it was noticed that the hour hand and minute hand had been interchanged and attached to the wrong axes. The result was that the hour hand moved with a speed twelve times higher than the minute hand. When the clock maker arrived, a remarkable thing happened: on the moment he inspected the clock, it showed exactly the right time again.

The Question: If the clock started at 6 o'clock in the correct position, then what was the first moment that it showed the correct time again?

A Hint: Let a pair of correct hands move together with the pair of wrong hands.

Answer: Suppose that a second pair of hands turns together with the wrong pair of hands, but then in the correct way. When the wrong pair is in the same position as the correct pair, this means that the time is shown in the right way. First look at the minute hands that are at the twelve. The "wrong" hand turns twelve times slower than the "correct" hand. Let x be the distance (in minutes) that the "wrong" hand has progressed when the two minute hands are in the same position again. The "correct" hand then has progressed 60+x minutes (one complete round more). So it then holds that 12x = 60+x. This means that x = 5 5/11 minutes.
For the hour hands that start at six holds the same. The confused clock therefore shows the correct time again after 60 + 5 5/11 minutes, so at 5 5/11 minutes past 7.