Eureka!

Toiling in an arcane area that totally baffles most ordinary mortals, mathematicians usually despair of even trying to explain their work to laymen. Yet recently two University of Illinois mathematicians announced a breakthrough of such widespread interest that even the reticent American Mathematical Society issued a rare press release. The news: after more than a century of futile brain racking, one of mathematics' most famous teasers--the so-called four-color conjecture--has finally been proved.

First stated in 1853 by a London graduate student named Francis Guthrie, the conjecture is simple. It says that no more than four colors are needed to shade any map so that no two adjoining countries are the same color. Though the experience of countless cartographers over the years supports the truth of this statement, mathematicians have never been able to prove it for all cases. Hence there remained the gnawing feeling that there just might be one instance where, say, five colors were needed instead of only four. Indeed, when Scientific American's puckish columnist Martin Gardner last year announced that such a "counter example" had indeed been found, it stunned math buffs everywhere--until they realized the claim was an April Fool's gag.

New Frontiers. The proof announced by Mathematicians Kenneth Appel and Wolfgang Haken in this month's math society Bulletin is no joke, however. They began by viewing the different possible maps that might be constructed in terms of simple and therefore mathematically manageable dots and lines. By this "graph" system, each country became a point; boundaries between countries became lines linking the dots. Painstakingly examining every imaginable map that could be fashioned out of these points and lines, Appel and Haken concluded that no matter how complex the map was, it had to contain at least one of 1,936 basic forms--or, in the jargon that helps keep mathematics mysterious, reducible configurations--that they had identified. Then they fed the forms into a computer and asked, in effect, whether all possible maps containing these configurations could indeed be made with only four colors. The electronic brain wrestled with the question for some 1,200 hours, during which it made some 10 billion separate, logical decisions. Finally the machine replied yes, and the four-color conjecture turned from theory into fact.

For mathematics, Appel and Haken's achievement may mean more than the end to a stubborn problem. Up to now, many theorists have been wary of using computers rather than simple, elegant blackboard equations to seek out basic mathematical truths; tedious chores like tracking a spacecraft, which involve no new principles, were left to the electronic brains. Now, by dramatically showing that there may be certain fundamental questions that only the high-speed electronic whizzes can answer, Appel and Haken may well have ushered in a new era of computer computation on the frontiers of higher mathematics.

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