The problem of counting the number of self-avoiding polygons on a square grid, - therbytheirperimeterortheirenclosedarea, is aproblemthatis soeasytostate that, at ?rst sight, it seems surprising that it hasn t been solved. It is however perhaps the simplest member of a large class of such problems that have resisted all attempts at their exact solution. These are all problems that are easy to state and look as if they should be solvable. They include percolation, in its various forms, the Ising model of ferromagnetism, polyomino enumeration, Potts models and many others. These models are of...