It is known that the four-colour problem for the faces of a map on a sphere is isomorphic with the four-colour problem for the vertices of its dual, and the problem is here discussed in the latter form. The isomorphs described below are concerned with codes for the four colours and for a change in colour as we pass along an edge from vertex to vertex. In the first (algebraic) isomorph, the coding involves the four fourth roots of unity, and leads to a graphical representation in the complex plane. In the second (arithmetical) isomorph, the coding involves the integers mod 4, and also the face-edge incidence matrix of the dual.