Completing partial latin squares: Cropper's question

Hall’s condition is a well-known necessary condition for the existence of a proper coloring of a graph from prescribed lists. Completing a partial latin square is a very special kind of graph list-coloring problem. Cropper’s question was: is Hall’s condition sufficient for the existence of a completion of a partial latin square? The folk belief that the answer must be no is confirmed here, but, also, six theorems giving necessary and sufficient conditions for completion of partial latin squares in different circumstances are recast in the form: when the prescribed cells in a partial latin square form such-and-such a configuration, then not only is Hall’s condition sufficient for completion, but, in each of these cases, a small subset of the large set of inequalities constituting Hall’s condition suffice.