We establish explicit relations between the residual entropy of the onedimensional Ising chain with a nearest-neighbour ferromagnetic and & th-neighbour antiferromagnetic interaction, and the residual entropies of the one-dimensional king chain with many-neighboured antiferromagnetic interactions in the corresponding maximum critical fields. The obtained results are, in particular, relevant to the chains that appear in the axial next-nearest-neighbour Ising model.The axial next-nearest-neighbour Ising (or ANNNI) model in d dimensions consists of (d -1)dimensional layers of spins with nearest-neighbour ferromagnetic coupling, Jo > 0, within layers but competing ferromagnetic, J 1 , and antiferromagnetic, J2 < 0,