A lesson in displacement

When an ice cube melts in a glass of water, does the level of the water fall or rise?

Neither. Archimedes' principle states that a body immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid. When the ice cube is floating, the buoyant force is equal to the weight of the ice cube, but since the buoyant force is equal to the weight of the water displaced by the submerged portion of the ice cube, the floating ice cube displaces its own weight in water. When the ice cube melts, it turns into its own weight in water, thereby leaving the overall water level unchanged.