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The crown made out of gold and silver has the same mass like the crown made out of pure gold - 1 kg. For this reason, Archimedes could not have found out if the crown was made out of pure gold with using scales only.

The volume of 1 kg of pure gold (left side on the picture) and the volume of 1 kg of the gold and silver mixture (right side of the picture) is different. Therefore, the volume of the complex body of the crown had to be measured. This was done simply by submerging it into a fluid.