"Look at the
In considering a few puzzles concerning clocks and watches, and the times
recorded by their hands under given conditions, it is well that a particular
convention should always be kept in mind. It is frequently the case that a
solution requires the assumption that the hands can actually record a time
involving a minute fraction of a second. Such a time, of course, cannot be
really indicated. Is the puzzle, therefore, impossible of solution? The
conclusion deduced from a logical syllogism depends for its truth on the two
premises assumed, and it is the same in mathematics. Certain things are
antecedently assumed, and the answer depends entirely on the truth of those
"If two horses," says Lagrange, "can pull a load of a certain weight, it is
natural to suppose that four horses could pull a load of double that weight, six
horses a load of three times that weight. Yet, strictly speaking, such is not
the case. For the
inference is based on the assumption that the four horses pull alike in amount
and direction, which in practice can scarcely ever be the case. It so happens
that we are frequently led in our reckonings to results which diverge widely
from reality. But the fault is not the fault of mathematics; for mathematics
always gives back to us exactly what we have put into it. The ratio was constant
according to that supposition. The result is founded upon that supposition. If
the supposition is false the result is necessarily false."
If one man can reap a field in six days, we say two men will reap it in three
days, and three men will do the work in two days. We here assume, as in the case
of Lagrange's horses, that all the men are exactly equally capable of work. But
we assume even more than this. For when three men get together they may waste
time in gossip or play; or, on the other hand, a spirit of rivalry may spur them
on to greater diligence. We may assume any conditions we like in a problem,
provided they be clearly expressed and understood, and the answer will be in
accordance with those conditions.
WAS THE TIME?
"I say, Rackbrane, what is the time?" an acquaintance asked our friend the
professor the other day. The answer was certainly curious.
"If you add one quarter of the time from noon till now to half the time from
now till noon to-morrow, you will get the time exactly."
What was the time of day when the professor spoke?
How many minutes is it until six o'clock if fifty minutes ago it was four
times as many minutes past three o'clock?
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?
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.
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 o'clock p.m. and midnight? And out of all the pairs
of times indicated by these changes, what is the exact time when the minute hand
will be nearest to the point IX?
One of the big clocks in the Cogitators' Club was found the other night to
have stopped just when, as will be seen in the illustration, the second hand was
exactly midway between the other two hands. One of the members proposed to some
of his friends that they should tell him the exact time when (if the clock had
not stopped) the
second hand would next again have been midway between the minute hand and the
hour hand. Can you find the correct time that it would happen?
We have here a stop-watch with three hands. The second hand, which travels
once round the face in a minute, is the one with the little ring at its end near
the centre. Our dial indicates the exact time when its owner stopped the watch.
You will notice that the three hands are nearly equidistant. The hour and minute
hands point to spots that are exactly a third of the circumference apart, but
the second hand is a little too advanced. An exact equidistance for the three
hands is not possible. Now, we want to know what the time will be when the three
hands are next at exactly the same distances as shown from one another. Can you
state the time?
On Friday, April 1, 1898, three new clocks were all set going precisely at
the same time—twelve noon. At noon on the following day it was found that clock
A had kept perfect time, that clock B had gained exactly one minute, and that
clock C had lost exactly one minute. Now, supposing that the clocks B and C had
not been regulated, but all three allowed to go on as they had begun, and that
they maintained the same rates of progress without stopping, on what date and at
what time of day would all three pairs of hands again point at the same moment
at twelve o'clock?
RAILWAY STATION CLOCK.
A clock hangs on the wall of a railway station, 71 ft. 9 in. long and 10 ft.
4 in. high. Those are the dimensions of the wall, not of the clock! While
waiting for a train we noticed that the hands of the clock were pointing in
opposite directions, and were parallel to one of the diagonals of the wall. What
was the exact time?
A facetious individual who was taking a long walk in the country came upon a
yokel sitting on a stile. As the gentleman was not quite sure of his road, he
thought he would make inquiries of the local inhabitant; but at the first glance
he jumped too hastily to the conclusion that he had dropped on the village
idiot. He therefore decided to test the fellow's intelligence by first putting
to him the simplest question he could think of, which was, "What day of the week
is this, my good man?" The following is the smart answer that he received:—
"When the day after to-morrow is yesterday, to-day will be as far from Sunday
as to-day was from Sunday when the day before yesterday was to-morrow."
Can the reader say what day of the week it was? It is pretty evident that the
countryman was not such a fool as he looked. The gentleman went on his road a
puzzled but a wiser man.