Using the Landauer formula, the conductance of short gold wires is studied. The required electronic structure calculations are performed with a self-consistent tight-binding method. We consider gold wires of single-atom diameter with a variable number (N=1,...,5) of atoms. Depending on N, we find considerable conductance variations with one conductance quantum being the upper limit. The results are confirmed by means of Friedel's sum rule. Tip-shaped clusters are used to provide the contact-wire interfaces and the relation between various tip structures and the conductance is discussed. Our predictions about the conductance variations agree qualitatively with new experimental results [R. H. M. Smit , Phys. Rev. Lett. 91, 076805-1 (2003)]. (C) 2004 American Institute of Physics.