The crystallographic phase problem is indeterminate in the absence of additional chemical information. A successful ab initio approach to the macromolecular phase problem must employ sufficient chemical constraints to limit the solutions to a manageably small number. Here we show that commonly employed chemical constraints - positivity, atomicity and a solvent boundary - leave the phase problem greatly underdetermined for Fourier data sets of moderate (2.5-3.0 A) resolution. Entropy maximization is also beset by multiple false solutions: electron-density maps are readily generated which satisfy the same Fourier amplitude constraints but have higher entropies than the true solution. We conclude that a successful ab initio approach must make use of high-resolution Fourier data and/or stronger chemical constraints. One such constraint is the connectivity of the macromolecule. We describe a rapid algorithm for measuring the connectivity of a map, and show its utility in reducing the multiplicity of solutions to the phase problem.