### Tip the Balance

Source: Gazette of the Australian Mathematical Society

Problem: A balance scale sits on the teacher’s table, currently tipped to the right. There is a set of weights on the scales, and on each weight is the name of at least one student. The teacher chooses a set of students to enter the classroom. One by one, each chosen student will move each weight carrying his or her name to the opposite side. Prove that the teacher can always choose a set of students to tip the scales to the left.

Update (June 17, 2010)
Solution: Posted by me in comments!

1. We will use the Averaging argument.

Consider all subsets of the set of students including the empty set and the full set. Each weight
will be on the left side of the balance half of the time. So, the total weight on the left for all these subsets is the same as total weight on the right side of the balance. Since the empty set results in a tip on the right, some other set must tip it on the left.

Hence, by averaging argument, there exists a set of students which the teacher can call to make
sure that the balance tips to the left.