Suppose that the value of a gem increases geometrically with the karat size. A 1 karat gem is worth $1,000; a 2 karat gem is worth $4,000; a 3 karat gem is worth $9,000. And so on. A man trades two gems of the same size for two gems of different sizes (no fractional karats). What are the smallest karat sizes that will result in an equal trade?

As an example, a 3-karat gem and 4-karat gem would be an equal trade for one 5-karat gem. But we are looking for two different-sized gems that add up the same value of two same-sized gems.