We study Ostwald ripening of two-dimensional adatom and advacancy islands on a crystal surface by means of kinetic Monte Carlo simulations. At large bond energies the islands are square-shaped, which qualitatively changes the coarsening kinetics. The Gibbs--Thomson chemical potential is violated: the coarsening proceeds through a sequence of `magic sizes corresponding to square or rectangular islands. The coarsening becomes attachment-limited, but Wagners asymptotic law is reached after a very long transient time. The unusual coarsening kinetics obtained in Monte Carlo simulations are well described by the Becker--Doring equations of nucleation kinetics. These equations can be applied to a wide range of coarsening problems.
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