THE FOUR ELOPEMENTS.
(
Measuring, Weight, and Packing Puzzles.)
Colonel B---- was a widower of a very taciturn disposition. His
treatment of his four daughters was unusually severe, almost cruel, and
they not unnaturally felt disposed to resent it. Being charming girls
with every virtue and many accomplishments, it is not surprising that
each had a fond admirer. But the father forbade the young men to call at
his house, intercepted all letters, and placed his daughters under
stricter supervision than ever. But love, which scorns locks and keys
and garden walls, was equal to the occasion, and the four youths
conspired together and planned a general elopement.
At the foot of the tennis lawn at the bottom of the garden ran the
silver Thames, and one night, after the four girls had been safely
conducted from a dormitory window to _terra firma_, they all crept
softly down to the bank of the river, where a small boat belonging to
the Colonel was moored. With this they proposed to cross to the opposite
side and make their way to a lane where conveyances were waiting to
carry them in their flight. Alas! here at the water's brink their
difficulties already began.
The young men were so extremely jealous that not one of them would allow
his prospective bride to remain at any time in the company of another
man, or men, unless he himself were present also. Now, the boat would
only hold two persons, though it could, of course, be rowed by one, and
it seemed impossible that the four couples would ever get across. But
midway in the stream was a small island, and this seemed to present a
way out of the difficulty, because a person or persons could be left
there while the boat was rowed back or to the opposite shore. If they
had been prepared for their difficulty they could have easily worked out
a solution to the little poser at any other time. But they were now so
hurried and excited in their flight that the confusion they soon got
into was exceedingly amusing--or would have been to any one except
themselves.
As a consequence they took twice as long and crossed the river twice as
often as was really necessary. Meanwhile, the Colonel, who was a very
light sleeper, thought he heard a splash of oars. He quickly raised the
alarm among his household, and the young ladies were found to be
missing. Somebody was sent to the police-station, and a number of
officers soon aided in the pursuit of the fugitives, who, in consequence
of that delay in crossing the river, were quickly overtaken. The four
girls returned sadly to their homes, and afterwards broke off their
engagements in disgust.
For a considerable time it was a mystery how the party of eight managed
to cross the river in that little boat without any girl being ever left
with a man, unless her betrothed was also present. The favourite method
is to take eight counters or pieces of cardboard and mark them A, B, C,
D, a, b, c, d, to represent the four men and their prospective brides,
and carry them from one side of a table to the other in a matchbox (to
represent the boat), a penny being placed in the middle of the table as
the island.
Readers are now asked to find the quickest method of getting the party
across the river. How many passages are necessary from land to land? By
"land" is understood either shore or island. Though the boat would not
necessarily call at the island every time of crossing, the possibility
of its doing so must be provided for. For example, it would not do for a
man to be alone in the boat (though it were understood that he intended
merely to cross from one bank to the opposite one) if there happened to
be a girl alone on the island other than the one to whom he was engaged.
Answer:
If there had been only three couples, the island might have been
dispensed with, but with four or more couples it is absolutely necessary
in order to cross under the conditions laid down. It can be done in
seventeen passages from land to land (though French mathematicians have
declared in their books that in such circumstances twenty-four are
needed), and it cannot be done in fewer. I will give one way. A, B, C,
and D are the young men, and a, b, c, and d are the girls to whom they
are respectively engaged. The three columns show the positions of the
different individuals on the lawn, the island, and the opposite shore
before starting and after each passage, while the asterisk indicates the
position of the boat on every occasion.
Lawn. | Island. | Shore.
| |
ABCDabcd | |
ABCD cd | | ab
ABCD bcd | | a
ABCD d | bc | a
ABCD cd | b | a
CD cd | b | AB a
BCD cd | b | A a
BCD | bcd | A a
BCD d | bc | A a
D d | bc | ABC a
D d | abc | ABC
D d | b | ABC a c
B D d | b | A C a c
d | b | ABCD a c
d | bc | ABCD a
d | | ABCD abc
cd | | ABCD ab
| | ABCD abcd
Having found the fewest possible passages, we should consider two other
points in deciding on the "quickest method": Which persons were the most
expert in handling the oars, and which method entails the fewest
possible delays in getting in and out of the boat? We have no data upon
which to decide the first point, though it is probable that, as the boat
belonged to the girls' household, they would be capable oarswomen. The
other point, however, is important, and in the solution I have given
(where the girls do 8-13ths of the rowing and A and D need not row at
all) there are only sixteen gettings-in and sixteen gettings-out. A man
and a girl are never in the boat together, and no man ever lands on the
island. There are other methods that require several more exchanges of
places.
Informational
