The Problem: You have twelve coins. One is counterfeit, but you do not know if it is too heavy or too light. What is the minimum number of weighings on a balance scale you need to identify the counterfeit coin, and to say whether it is too heavy or too light?

The Solution: 3 weighings suffice. There are 12(2) = 24 possibilities for the counterfeit coin (e.g., coin 1 is too heavy, coin 1 is too light, coin 2 is too heavy, etc.). In each weighing there are 3 possible outcomes (left side goes down, right side goes down, or it balances). Since 3^2 = 9 < 24, clearly 2 weighings will not suffice. To see that 3 will suffice, scroll down to consider the following interactive algorithm. You might want to think some more about how to do it in 3 weighings before you see the way I did it. (My algorithm is certainly not unique, but it works.)

Choose a counterfeit coin, from 1-12, and decide whether it's heavy or light. After three weighings, I will identify your coin and its defect.

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