Why Are So Many Problems Unsolved?

Abstract

The problems discussed in this book, particularly that of counting the number of polygons and polyominoes in two dimensions, either by perimeter or area, seems so simple to state that it seems surprising that they haven't been exactly solved. The counting problem is so simple in concept that it can be fully explained to any schoolchild, yet it seems impossible to solve. In this chapter we develop what is essentially a numerical method that provides, at worst, strong evidence that a problem has no solution within a large class of functions, including algebraic, differentiably finite (D-finite) [27, 26] and at least a sub-class [7] of differentiably algebraic functions, called constructible di..