Authorâ€™s abstract: Despite the advice of his advisor that he choose a more modest problem to solve for his dissertation, a young graduate student at the University of Wisconsin has succeeded in explaining a particularly perplexing and deep pattern in a numerical sequence. The sequence represents the number of ways in which positive integers can be expressed as the sums of other positive integers. Not counting the years of study that prepared Karl Mahlburg for his chosen problem, exploring, conjecturing, and proving theorems for the problem took about one and a half years of intense concentration. He finally succeeded in the spring of 2005, and his work is now attracting a lot of attention in the mathematical community. His methods and results are likely to have an impact on other problems in both pure and applied areas of mathematics.