Since the time of Boltzmann, one of the great mysteries in physics has been the apparent mismatch between the time-reversal symmetry of the laws of nature and the irrerversible growth of entropy expressed by the second law of thermodynamics. It is widely believed that special initial conditions must be imposed on any time-symmetric law if its solutions are to exhibit behaviour of any kind that defines an ‘arrow of time’. In a recent paper in Physical Review Letters (arXiv:1409.0917), collaborators and I have shown this is not so. My talk will be based on this paper, in which we study the simplest non-trivial time-symmetric gravitational law that can be used to model a dynamically closed universe. Because of specific properties of this law, its typical solutions all divide at a uniquely defined point into two halves. In each a well-defined measure of shape complexity fluctuates but grows irreversibly between rising bounds from that point. Each solution can be viewed as having a single past and two distinct futures emerging from it. Any internal observer must be in one half of the solution and from observations within it will find an arrow of time despite the underlying time-symmetry of the law. We are still far from a complete explanation of the second law, but this seems to be a promising first step.