We consider ``magnetized brane'' compactifications of the type I/heterotic string on K3 with U(1) background fluxes. The gauge group and matter content of the resulting six-dimensional vacua are parameterized by a matrix encoding a lattice contained within the even, self-dual lattice Γ3,19. Mathematical results of Nikulin on lattice embeddings make possible a simple classification of all such solutions. We find that every six-dimensional theory parameterized in this way by a negative semi-definite matrix whose trace satisfies a simple tadpole constraint can be realized as a K3 compactification. This approach makes it possible to explicitly and efficiently construct all models in this class with any particular allowed gauge group and matter content, so that one can immediately ``dial-a-model'' with desired properties.