Back to the Future: 100-Year-Old AMM Problem May Have Been Earliest Hint of Percolation Theory

August 25, 2010

A little-known problem that appeared in the first issue of American Mathematical Monthly (March, 1894, p. 99) may have been the first reference in mathematical history to what has become a 21st-century subject in mathematics: percolation theory.

According to probabilist Harry Keston (Cornell University), Stanislav Smirnov's decade-long work in percolation theory had put it on a solid mathematical foundation—and it resulted in Smirnov winning a 2010 Fields Medal.

De Volson Wood (Stevens Institute of Technology) proposed the apparently simple AMM problem, saying that an "actual case suggested the following":

"An equal number of white and black balls of equal size are thrown into a rectangular box, what is the probability that there will be contiguous contact of white balls from one end of the box to the opposite end? As a special example, suppose there are 30 balls in the length of the box, 10 in the width, and 5 (or 10) layers deep."

P.H. Philbrick of Lake Charles, Louisiana, sent in a solution to the problem, to which the Monthly’s Problem Editor wrote in the June 1894 issue was “not entirely satisfactory.” The Editor continued, writing, "The problem is a pretty good one and if anyone will furnish a complete solution to it, we will publish it in the next issue."

The Monthly is still waiting.

For more, see the Math in the News item "Studying Percolation on Infinite Graphs" (May 28, 2008).

Source: +plus Magazine (August 20, 2010)