The globally minimum energy configurations of simpleHPlatticemodels (which use only two amino acid types, positioned on the vertices of a square lattice) of proteins have been established for short sequences. Here we investigate the folding of such proteins to this globally minimum energy configuration, both cotranslationally (as they are manufactured, sequentially, in the ribosome) and starting from a fully extended state. In order to do this we model the folding process and develop a heuristic method for finding local energy minima. Two main results emerge. First, some sequences do fold better cotranslationally than from a fully extended state and second, this can be due to cotranslational folding leading to an initial local energy minimum from which movement to the global minimum is efficient. Sequences for which this is true tend to have a higher density of hydrophobic residues at the start than at the finish. Structural properties of sequences that fold better cotranslationally than from a fully extended state are also identified.