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Amusements in Mathematics

58.—A TIME PUZZLE.
How many minutes is it until six o'clock if fifty minutes ago it was four times as many
minutes past three o'clock?
59.—A PUZZLING WATCH.
A friend pulled out his watch and said, "This watch of mine does not keep perfect time; I
must have it seen to. I have noticed that the minute hand and the hour hand are exactly
together every sixty-five minutes." Does that watch gain or lose, and how much per hour?
60.—THE WAPSHAW'S WHARF MYSTERY.
There was a great commotion in Lower Thames Street on the morning of January 12,
1887. When the early members of the staff arrived at Wapshaw's Wharf they found that
the safe had been broken open, a considerable sum of money removed, and the offices
left in great disorder. The night watchman was nowhere to be found, but nobody who had
been acquainted with him for one moment suspected him to be guilty of the robbery. In
this belief the proprietors were confirmed when, later in the day, they were informed that
the poor fellow's body had been picked up by the River Police. Certain marks of violence
pointed to the fact that he had been brutally attacked and thrown into the river. A watch
found in his pocket had stopped, as is invariably the case in such circumstances, and this
was a valuable clue to the time of the outrage. But a very stupid officer (and we
invariably find one or two stupid individuals in the most intelligent bodies of men) had
actually amused himself by turning the hands round and round, trying to set the watch
going again. After he had been severely reprimanded for this serious indiscretion, he was
asked whether he could remember the time that was indicated by the watch when found.
He replied that he could not, but he recollected that the hour hand and minute hand were
exactly together, one above the other, and the second hand had just passed the forty-ninth
second. More than this he could not remember.
What was the exact time at which the watchman's watch stopped? The watch is, of
course, assumed to have been an accurate one.
61.—CHANGING PLACES.
The above clock face indicates a little before 42 minutes past 4. The hands will again
point at exactly the same spots a little after 23 minutes past 8. In fact, the hands will have
changed places. How many times do the hands of a clock change places between three
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