Answer:

This is the Archimedes Principle, using density to determine the purity of the object.  Density is mass/volume.  The problem for the king is he could melt the crown and see if there are any impurites that come out, but this would destroy the crown.  However, if he decided to melt the crown and impurities came out during the melting process the king would know he was cheated. Even if the mass of metal is all fake, he would be able to determine how much metal (volume) he really has now that it is melted into a brick and he could weigh it.  If the weight is a different weight than an equal volume of gold than the king knows he has been cheated.

Archimedes figured out a way to do this without melting the crown.  He reasoned that he could determine the volume of metal of the crown by submerging it in water.  The volume of water displaced is the volume of metal.  This volume of metal, if it is the same gold the king commissioned, should have the same weight as an equal volume of the original gold.  Thus, by weighing the crown and dividing it by its volume the density can be determined.  A sample of the commissioned gold is then weighed and its volume is determined to get its density. If the two densities are the same the craftsman was honest.


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