Archemedes'+Principle

Out of the water both the crown and gold bar balance, so have the same mass. If the water level rides equally for both the crown and gold bar and the balance stays level when they are submerged, then the crown and gold bar have the same density. If they have the same density they are both gold. The buoyant force will equal the mass of the fluid displaced by an object (Archimedes' Principle). Density equals mass divided by volume, so density times volume will equal the mass. So the buoyant force divided by volume will equal density. Buoyant Force = Density x Volume

The crown is gold: If two objects balance out of the water they have the same mass. (Mass of object 1 = Mass of object 2) If two submerged objects have the same volume they will have the same buoyant force. (Volume of object 1 = Volume of object 2) and (buoyant force of object 1 = buoyant force of object 2) If both objects have the same mass and volume, then they will have the same density. (Density of object 1 = Density of object 2) Two objects with the same density and volume are most likely made of the same substance.

Demonstration of changing the amount of displaced fluid.