It's deceptively easy to understand the problem, but it stumped the experts for decades. Now two mathematicians have solved a sticky little puzzle that has furrowed brows all over the world since the 1950s.
In 1957, Polish mathematician Hugo Steinhaus challenged his colleagues to find a curious set of numbers. First, imagine a set of numbers arrayed in a regular grid, like intersections within an infinite piece of graph paper. Now imagine another set of numbers, dusted in an irregular pattern onto a plane. Steinhaus asked if it were possible to make a set such that no matter how you placed the grid on the plane, the grid and the irregular set always intersect at exactly one point. "It's kind of a crazy problem," says Stephen Jackson, a mathematician at the University of North Texas, Denton. "Part of its appeal is that it's an elementary question."
Yet it had eluded solution, at least until Jackson and his co-worker, R. Daniel Mauldin, tackled the problem. In an article in the current issue of the Proceedings of the National Academy of Sciences, Jackson and Mauldin show how they combined techniques from several branches of mathematics--number theory, which deals with the basic properties of numbers; set theory, which describes collections of objects; and geometry, which deals with the relation of objects in space--to answer Steinhaus's problem in the affirmative. Yes, there is such a set.
"It's a really deep piece of work," says Jan Mycielski, a mathematician at the University of Colorado, Boulder. But although the mathematicians were able to prove that the set exists, they are just beginning to figure out its properties. Topologically, the set is rather ugly and irregular, making it all but impossible to visualize. But the proof itself might yield clues about how to solve several other, related, nagging problems, such as a handful that deal with assigning colors to points on the plane. "Hopefully, there are some ideas here that will be useful somewhere else," says Jackson.
Biography of Hugo Steinhaus