| | Ivars Peterson's MathLand |
August 19, 1996
The showman comes to a river and finds a small boat that holds only himself and one passenger. For obvious reasons, he can't leave the wolf alone with the goat, or the goat with the cabbages. How does he get his cargo safely to the other side?
Brainteasers that involve ferrying people and their belongings across a river under trying circumstances have been around for centuries. This particular version dates back to the eighth century and the writings of Alcuin, a poet, educator, cleric, and friend of Charlemagne.
In the sixteenth century, a more elaborate edition of this problem, proposed by Venetian mathematician Niccolo Fontana Tartaglia, featured three beautiful brides and their young, handsome, and intensely jealous husbands, who come to a river. The small boat that is to take them across holds no more than two people. To avoid any 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 them all across the river -- without igniting an angry outburst?
It turns out that 11 trips are required. Five passages are needed for just two couples. With four or more couples, however, it's impossible to accomplish the crossings under the required conditions.
A nineteenth-century version has three missionaries and three cannibals together on one side of a river, with a boat that holds only two people. In this case, the cannibals must never be allowed to outnumber missionaries on either bank. It takes nine trips to get everyone across.
Here's another variant of the crossing problem, this time from a Russian source: Three soldiers have to cross a river without a bridge. Two boys with a boat agree to help the soldiers, 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. Given that none of the soldiers can swim, it would seem that in these circumstances just one soldier could cross the river. Yet all three soldiers eventually end up on the other bank and return the boat to the boys. How do they do it?
Now, what happens if, in each of these problems, there's an island in the middle of the river, which can be used as a temporary landing place? In some cases, the island makes no difference in the total number of trips or in the feasibility of a successful transfer, and sometimes it actually reduces the number of crossings necessary.
These puzzles and their many variants represent ways of dressing up relatively straightforward mathematical problems. They invite attempts at solution that range from trial and error to extensive mathematical analysis. One can even try to model a given problem with slips of paper that represent passengers and boat.
Mathematician David Singmaster of the Polytechnic of the South Bank in London, England, has compiled an extensive bibliography of material devoted to recreational mathematics. "Recreational problems act as historical markers, showing the transmission of mathematics (and culture in general) in time and space," he notes. "In particular, they illustrate the fact that most of the more algebraic and arithmetic parts of mathematics have their origins in the Orient, beginning with Babylonia and China and being transmitted through India and the Arabs.
"One must be a little careful with some of these problems, as past cultures were often blatantly sexist or racist," Singmaster warns. "But such problems also show what the culture was like.... The river crossing problem of the jealous husbands is quite sexist and transforms into masters and servants, which is classist, then into missionaries and cannibals, which is racist. With such problems, you can offend everybody!"
From the days of ancient Egypt to modern times, it's all done for the sake of turning what appear to be routine mathematical exercises into problems that tickle -- even challenge -- the mind.
Copyright © 1996 by Ivars Peterson.
Ball, W.W. Rouse, and H.S.M. Coxeter. 1974. Mathematical Recreations and Essays. Toronto: University of Toronto Press.
Krasner, Edward, and James R. Newman. 1989. Mathematics and the Imagination. Redmond, Wash.: Microsoft Press.
Brandreth, Gyles. 1985. Classic Puzzles. New York: Harper & Row.
Guy, Richard K., and Robert E. Woodrow, eds. 1994. The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History. Washington, D.C.: Mathematical Association of America. Perelman, Y.I. 1984. Fun with Maths and Physics. Moscow: Mir Publishers.
Peterson, Ivars. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman.
______. 1986. Games Mathematicians Play. Science News 130(Sept. 20):186-189.
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Ivars Peterson is the mathematics and physics writer at Science News (http://www.sciencenews.org/). He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, and Fatal Defect. His current work in progress is Adventures in Mathland: The Jungles of Randomness (to be published in 1997 by Wiley).