Weighty Matters

From the Second All Soviet Union Mathematical Competition, Leningrad 1968:

On a teacher’s desk sits a balance scale, on which are a set of weights. On each weight 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’s some set of students that you can let into the room that will will tip the scale to the left.

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Make a list of all possible subsets of the students, including the empty set and the full set, and suppose you let in all of these groups. Each weight would spend half its time in the left pan and half in the right, so the total weight for each pan is the same. Since the empty set produces a tip to the right, there must be some other set that produces a tip to the left.

October 16, 2014October 26, 2014 | Puzzles