PORTER JOHNSON
ELEMENTARY MATHEMATICS SMILE A
21 MARCH 1995

WEIGHING PROBLEM

One is given a set of 12 coins, one of which is counterfeit and either lighter or heavier than the other 11. We are allowed to use a [crude] balance, which can only tell us whether the coins on the left pan weigh less than, the same as, or more than the coins on the right pan. Devise a scheme which involves only three different weighings to determine which coin is counterfeit, and whether it is lighter or heavier than the others.

Remark: To solve this problem one must make full use of the three possible results of weighing, varying which and how many coins are on each side, to develop the scheme. I have found a scheme which works, which I will describe here. It may not be unique.

Let us begin by numbering the coins 1 through 12. [It is important that the coins be distinguishable, in spite of having the same weight--perhaps the dates on them are all different.

Weighing #1: Put coins 1, 2, 3, 4 on the left pan and coins 5, 6, 7, 8 on the right pan.