The fraudulent goldsmith.

Apparently Archimedes   was capable of strong mental concentration, and tales are told of his obliviousness to surroundings when engrossed by a problem. Typical is the frequently told story of King Hiero's crown and the suspected goldsmith.

It seems that King Hiero, desiring a crown of gold, gave a certain weight of the metal to a goldsmith, along with instructions. In due time the crown was completed and given to the king. Though the crown was of the proper weight, for some reason the king suspected that the goldsmith had pocketed some of the precious metal and replaced it with silver. The king didn't want to break the crown open to discover if it contained any hidden silver, and so in his perplexity he referred the matter to Archimedes. For a while, even Archimedes was puzzled. Then, one day when in the public baths, Archimedes hit upon the solution by discovering the first law of hydrostatics. In his flush of excitement, forgetting to clothe himself, he rose from his bath and ran home through the streets shouting, "Eureka, eureka" ("I have found it, I have found it").

The famous first law of hydrostatics appeared later as Proposition 7 of the first book of Archimedes' work  On Floating Bodies.

This law, which today every student of physics learns in high school, says that " a body immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid." This means that of two equal masses of different materials, that one having the greater volume will lose more when the two masses are weighed under water. Thus, since silver is more bulky than gold, it suffers a greater change when weighed under water than does an equal mass of gold. So all Archimedes had to do was to put the crown on one pan of a balance and an equal weight of gold on the other pan, and then immerse the whole in water. In this situation the gold would outweigh the crown if the latter contained any hidden silver. Tradition says that the pan containing the crown rose, and in this way the goldsmith was shown to be dishonest.