  Norine's antipodal-colouring conjecture, in a form given by Feder and Subi,
asserts that whenever the edges of the discrete cube are 2-coloured there must
exist a path between two opposite vertices along which there is at most one
colour change. The best bound to date was that there must exist such a path
with at most $n/2$ colour changes. Our aim in this note is to improve this
upper bound to $(\frac{3}{8}+o(1))n$.
