We report the first observation, by computer simulation, of a purely entropic demixing transition in a three-dimensional binary hard-core mixture. This transition is observed in a mixture of large and small cubes. We also find evidence for demixing in other hard-core fluids and, in the case of an athermal polymer solution, we observe a purely entropy-driven polymer collapse. For the study of both the hard-core demixing and polymer collapse, it was essential to use novel collective Monte Carlo moves.