Source: Australian Mathematical Society Gazette Puzzle Corner 35
There are some weights on the two sides of a balance scale. The mass of each weight is an integer number of grams, but no two weights on the same side of the scale share the same mass. At the moment, the scale is perfectly balanced, with each side weighing a total
of W grams. Suppose W is less than the number of weights on the left multiplied by the number of weights on the right.
Is it always true that we can remove some, but not all, of the weights from each side and still keep the two sides balanced?