The solar neutrino puzzle is deepening into a paradox that refutes the basic logic of the reaction chain that powers the sun by the fusion of protons into heavy elements. Experiments now reveal a serious anomaly in the relative neutrino fluxes from the different steps in the chain. Neutrinos from boron-8 at the end of the chain are seen but hardly any are seen from beryllium-7, without which the observed boron-8 cannot be made. The only apparent way to avoid a paradoxical "missing link" in the sun’s energy chain is a nonzero neutrino mass, an idea that can be tested in future experiments.