Games mathematicians play; mathematical puzzles, games and other pastimes both enlighten and amuse - includes related article on Eugene Strens Recreational Mathematics Collection at University of Calgary

Science News, Sept 20, 1986 by Ivars Peterson

Games Mathematicians Play

Have you heard the one about an itinerant entertainer traveling with a wolf, a goat and a basket of cabbages? The showman comes to a river and finds a small boat that holds only him 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 safety to the other side?

This well-known brainteaser has been around for centuries. In a version initiated in the 13th century, the puzzle involved three handsome young men who have three beautiful ladies for wives. All six are jealous of their spouses. Using a two-person boat, how many trips does it take to ferry them all across a river -- without igniting a fit of illicit passion?

Both of these problems and many other variations on this theme are simply ways of dressing up a relatively straight-forward mathematical problem. Since the days of the ancient Egyptians and Babylonians, such devices have often been used to turn a routine mathematical exercise into something that tickles and challenges the mind.

Mathematical puzzles and games are still remarkably popular. Throughout the world, puzzle addicts snap up many of the hundreds of such books published every year. Numerous magazines feature puzzle columns. Furthermore, the appearance of a new and ingenious puzzle can stir up frenzied activity. In three years, for instance, sales of Rubik's cubes exceeded 100 million.

Amusement is one of humankind's strongest motivating forces, says mathematician David Singmaster of the Polytechnic of the South Bank in London, England. Recreational problems, he adds, have spawned many mathematical fields. The origin of probability theory in questions about gambling is just one example.

Moreover, says Singmaster, an interesting problem is worth hours of lecturing. "Recreational mathematics offers a range of problems [that] have fascinated students for generations and should continue to do so for future generations."

Singmaster was one of about 100 people, a mixture of professional and amateur mathematicians, who recently spent a merry week at a conference on recreational and intuitive mathematics at the University of Calgary in Alberta. At this meeting, work was play and play was work. Puns punctuated lectures. Pen or pencil scribbles marked progress toward new puzzles or novel solutions for old ones. Participants wrestled with numbers, tiles, wooden blocks, counters, matchsticks, coins, cards and fiendishly interlocked wire rings.

Even cookies made an appearance. They play a tantalizing role in a new game introduced by mathematician James Propp of the University of California at Berkeley. He describes the game as follows: Imagine two children who take turns stealing cookies from a larder, each taking a single cookie every other day. Some of the cookies may go bad while sitting on the shelf, but fortunately each cookie is frosted with its own expiration date. Once that date is reached, the children avoid the spoiled cookie.

"The goal of each of these mean-spirited children," says Propp, "is...to have the spiteful pleasure of getting the last cookie." If no cookies spoil during the game, the game turns out to be very boring. The winner's identity then depends only on whether the number of cookies is odd or even. But when some cookies go bad, the game's outcome is much less predictable.

Propp taught his brother how to play the game and promptly lost to him. "He played randomly and beat me," Propp says. Now Propp is trying to analyze the game to see if there are strategies that guarantee a win.

In his analysis, each cookie is represented by a pile of counters, where a pile's height equals the number of days remaining in the cookie's life. On any given turn, the player takes away one of the piles, and one counter is removed from all of the remaining piles. The player who takes away the last pile is the winner.

Although Propp has uncovered some interesting patterns that apply to certain sets of counters, he has not yet found a general strategy that can be used for any starting group of cookies. "I have more than enough data," he says, "but a lack of conjectures."

Heiko Harborth of the Technische Universitat Braunschweig in West Germany plays with matches. "Matchsticks are among the cheapest and simplest objects for puzzles," he says. "Whole books have been devoted to matchstick puzzles." At the meeting, to keep his audience fully occupied, Harborth handed out boxes of matches to anyone who preferred working on a puzzle to listening to his lecture.

One group of matchstick (or toothpick) problems involves constructing patterns in which a given number of sticks meet end to end (without crossing each other) at every point in a geometric figure on a flat surface. For example, a figure made up of three sticks laid out as an equal-sided triangle has two sticks meeting at each corner. This is the smallest number of sticks that can be used to create a pattern in which two sticks meet at every vertex.