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.