Skipping stones across water has been a popular pastime for thousands of years - the rules of the game have remained unchanged since the time of the ancient Greeks - and the world record, set by J. Coleman-McGhee in 1992, is believed to be 38 rebounds. Following earlier attempts to analyse the physics of this ancestral human activity, we focus here on the crucial moment in stone skipping: when the stone bounces on the water's surface. By monitoring the collision of a spinning disc with water, we have discovered that an angle of about 20 degrees between the stone and the water's surface is optimal with respect to the throwing conditions and yields the maximum possible number of bounces.