On a teacher’s desk sits a balance scale, on which there is a set of weights.
On each weight there is the name of at least one student. As each student enters the classroom, she moves all the weights that bear her name to the other side of the scale.
Before any students enter, the scale is tipped to the right.
Prove that there is some set of students that you can let into the room that will tip the scale to the left.
Source: Soviet Union Mathematical Competition
Well, this certainly works for two students, who I will call A and B. There could be up to three weights, with names A, B, and AB. If any of those weights are not part of the set, imagine that there is a virtual weight which weighs zero pounds.
We can always get the two heaviest weights on the left side of the scale, by letting appropriate students into the room. And this is guaranteed to get the scale to tilt left.
If A is the lightest weight, invite only B into the room.
If B is the lightest weight, invite only A into the room.
If AB is the lightest weight, invite both A and B into the room.
So, n = 2 works.
If n = 3, there are 2^3 - 1 = 7 weights (some of which may be virtual) and 7 different combinations of names which can be called into the room. I need to give this some more thought.