As mathematicians often do, Euler took the fun out of the game. First he turned the problem into a diagram of networks (lines representing the paths) and nodes (dots representing the land masses). An odd node had an odd number of lines jutting out from it; an even node had an even number of lines. Euler then showed that it was impossible to walk in a continuous circuit -- without retracing your steps -- unless the diagram had no odd nodes or two odd nodes. Since the Konigsberg Bridge Problem had four odd nodes, it proved impossible.
While sitting on a bench, Rockmore tells the story of the 20th-century Hungarian mathematician George Polya, who was in Zurich for some years. Meandering through a park one afternoon, Polya kept running into a colleague and his girlfriend. The colleague believed that Polya was making contact on purpose. Perhaps eager to prove that he wasn't hitting on his friend's girl, Polya devised the Random Walk Problem. He eventually published a mathematical proof showing that if you walk around enough in an infinite grid, you will return to the same points over and over.
Gesturing toward a clump of seven trees in Central Park, Rockmore also returns to certain points. Seven, of course, is a prime. And primes remind Rockmore of Riemann and the seven Millennium Problems.
A Russian math whiz named Grigori Perelman posted a solution in 2002 to one of the problems -- the Poincare Conjecture. The conjecture, crafted in the early 1900s, asks if the properties of the two-dimensional surface of a sphere behave in the same way as the properties of a three-dimensional surface of a sphere, which is something we cannot see but can only imagine.
James Carlson, president of the institute, says that after two years of scrutiny by the professional community, Perelman's proof "still looks good." The institute's board of advisers will have to agree before he gets the money.
The seven problems are incredibly dense. Rockmore says he doesn't completely understand the Yang-Mills Theory, which is the mathematical theory underlying quantum physics. Another of the problems -- the Navier-Stokes Equations that would explain the ways that fluids flow -- is "not well defined," Rockmore says.
Readable explanations of the problems can be found in Devlin's book, "The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time. " The problems are also listed on the Clay Mathematics Institute Web site, http://www.claymath.org .
Some of the problems, Rockmore says, may be impossible to solve. "Maybe the Riemann Hypothesis turns out to be wrong," Rockmore says. "Some poor schmo may prove that. But he wouldn't get the million dollars."
Addition and Subtraction
There is much about mathematics that is unclear. For centuries, theoretical mathematics coexisted with uncertainty. "The literature is filled with incorrect theorems in the early 19th century," Rockmore says, but later in the century mathematicians began to insist on more precision.
Since then there has been more pressure to be precise, even as the world has become harder and harder to explain. But the occasional imprecision of mathematics remains a constant. "The zeitgeist affects mathematics," Rockmore says.
And mathematics affects the zeitgeist. Computers have given us the puffed-up notion that we can quantify just about anything -- even love. Rockmore points to eHarmony.com, the online matching service that relies on 29-question surveys to pair compatible people. As the Web site explains it: "The eHarmony.com compatibility matching models were created using factor analyses, multiple regression and discriminant analyses on data gathered from married couples." In other words, feelings are distilled to factors; attraction to analysis.
Back at the restaurant, you and Dan were given three pieces of bread. They brought to mind the Mathematics of Guilt, as described by British mathematician Rob Eastaway in "Why Do Buses Come in Threes? The Hidden Mathematics of Everyday Life." In Eastaway's example, the vicar's wife invites five friends over for tea. She offers her guests a plate of biscuits -- four are chocolate and one plain. All the guests like chocolate biscuits. The first guest takes a chocolate biscuit. So do the second and third. The fourth person knows that if she chooses the last chocolate biscuit, the fifth person will be forced to eat a plain one. She feels guilty and so doesn't take either one. The question, Eastaway asks, is: Should the other guests also have felt guilty, and if so, how do you decide mathematically the amount of guilt each guest should feel?
But guilt is not always based on taking the last biscuit. And love is not always founded on compatibility. Even the piece of cake that two people share might have a blemish on one half or a frayed corner on the other.
In the end, Rockmore decides not to have dessert after all. He orders a cup of coffee instead. Sometimes he stays up all night, trying to solve a problem.
