Brittle polycrystalline materials such as rocks, ceramics, and certain metals contain microcracks that can grow and coalesce under sufficiently high stress, resulting in failure and, possibly, fragmentation. Such processes are idealized in this paper by treating the cracks as circular disks whose coalescence forms clusters and can terminate growth after a number of intersections. A non-linear recurrence relation for the probability of cluster size is developed and solved by means of a generating function, providing information on the mean size of the crack clusters and the standard deviation. This solution leads to a simple expression for the percolation threshold. The probability of clusters of size n is also determined. Above the percolation threshold the probabilities of finite and infinite clusters are treated separately. Explicit expressions for the probabilities can be approximated by taking a Laplace transform in a simple case, thus clarifying the behaviour of the solution. Appendices show the relation of the theory to practical problems, Monte Carlo approaches, the probability of infinite clusters, and discuss the uniqueness of solutions to such geometrical problems, i.e. Bertrand's Paradox. Copyright © 2005 John Wiley & Sons, Ltd.