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Amusements in Mathematics

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?
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?