We address the question of determining the eigenvalues $\lambda_n$ (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp \ie, for which there exists an associated eigenfunction with $n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues.