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MatheMUSEments
Tricky Crossings
By Ivars Peterson
Muse, September 2004, p. 34-35.
Have you ever heard the one about a farmer traveling with a fox, a goose,
and a bag of grain?
The farmer comes to a river and finds a small boat that will hold
only two passengers. For obvious reasons, he can't leave the fox alone with the goose,
or the goose with the grain. How does he get his cargo safely to the other side?
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| Eric Roell | Brainteasers that involve ferrying people and their belongings across
a river under trying conditions have been around for centuries. This particular version
dates back more than 1100 years. Such puzzles are simply ways of dressing up fairly
standard math problems. To solve them, you can use math and logic, or you can try trial
and error. There are often lots of different ways to solve a given problem.
So, what does the farmer do? Here's one answer. The farmer crosses the river
with the goose and leaves the goose on the far shore, then returns alone to the near
shore. He brings the fox to the far shore and brings the goose back to the near shore.
He then brings the grain to the far shore and returns alone to the near shore. Finally,
the farmer brings the goose to the far shore. It takes seven trips.
Another type of river-crossing puzzle also features a predator, its prey,
and some food. In this case, however, the human can transport two items at one time, which
changes the logical structure of the problem. For instance, a herdsman in North Africa
might have to cross a river with a jackal (a predator), a goat (potential prey), and
fig leaves (potential snack for goat). Answer.
But there are countless other variations on the river-crossing problem
as well. Often the players and the situation reflect the culture of a particular place and
time. What is a fox in Europe becomes a jackal in North Africa.
Here's a Russian version. Two boys with a boat agree to help three soldiers
cross a river without a bridge. But the boat is so small it can support only one soldier
or two boys. A soldier and a boy can't be in the boat at the same time for fear of sinking it.
How many trips does it take to ferry all the soldiers across? Answer.
This cultural coloring also means old problems can be sexist or racist by
today's standards. In the 16th century, one version featured three beautiful brides and their
young, handsome, and intensely jealous husbands. The small boat that is to take them across
the river holds no more than two people. To avoid compromising situations, the crossings
must be arranged so that no woman is left with a man unless her husband is also present.
How many trips does it take to ferry all six across the river without an angry outburst? Answer.
A 19th-century version features three missionaries, three cannibals, and a
boat that holds only two people. In this case, the cannibals must never be allowed to outnumber
missionaries on either bank. How many trips does it take?
Answer.
New versions of the river-crossing problem continue to be invented. One puzzle found
on the Internet involves crossing a rickety bridge at night within a certain time period with just
one lantern to light the way. See
http://www.sciencenewsforkids.org/articles/20040630/PuzzleZone.asp.
Even job seekers might find they have to solve one of these puzzles. They're
sometimes part of the interview at places such as Microsoft. Take about a pop quiz!
From the days of ancient Egypt to modern times, river-crossing problems have
turned routine math exercises into puzzles that tickle your mind.
Answers
The Jackal, the Goat, and the Fig Leaves. One solution is to
take the jackal and the goat, leave the jackal while returning with the goat, then ferry across the
goat and the fig leaves. Three trips. Back.
The Russian Soldiers and the Boys. Both boys go to the opposite bank;
one of them brings the boat back to the soldiers; a soldier crosses the river; the boat returns with
the other boy; both boys cross the river; one boy returns with the boat;
the second soldier crosses the river; the second boy returns with the boat; both boys cross the
river; one boy returns with the boat; the third soldier crosses the river; and the second boy
returns with the boat. Twelve trips. Back.
The Three Beautiful Brides and Their Jealous Husbands. Say the couples
are named John and Joan, Edward and Elinor, Richard and Rose. Elinor and Rose cross; Rose returns;
Joan and Rose cross; Rose returns; John and Edward cross; John and Joan return; Richard and John cross;
Elinor returns; Elinor and Joan cross; Elinor returns; Rose and Elinor cross. Eleven trips. Back.
The Cannibals and the Missionaries. A missionary and a
cannibal cross; the missionary returns; two cannibals cross; one cannibal returns; two
missionaries cross; one missionary and one cannibal return; two missionaries cross; one
cannibal returns; two cannibals cross; one cannibal returns; and the remaining two cannibals cross. Eleven trips. Back.
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