AbstractThe following problem was originally posed by B. H. Neumann and H. Neumann. Suppose that a group G can be generated by n elements and that H is a homomorphic image of G. Does there exist, for every generating n‐tuple of H, a homomorphism and a generating n‐tuple of G such that ?M. J. Dunwoody gave a negative answer to this question, by means of a carefully engineered construction of an explicit pair of soluble groups. Via a new approach we produce, for , infinitely many pairs of groups that are negative examples to Neumanns' problem. These new examples are easily described: G is a free product of two suitable finite cyclic groups, such as , and H is a suitable finite projective special linear group, such as for a prime . A small modification yields the first negative examples with H infinite.