Balancing Act Puzzle
Source: Australian Mathematical Society Gazette Puzzle Corner 35 Problem: 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? Solution: Highlight the part between the * symbols for the answer. * Yes - in fact something stronger holds: some single weight sits on BOTH pans, so removing that one weight from each side keeps the balance. Proof (Govind's): let the left pan hold m weights a_1 < ... < a_m and the right pan n weights b_1 < ... < b_n, with both sums equal to W < mn. Suppose no weight value appears on both ...