You have 10 bags of gold coins. Each bag contains 10 gold coins. In nine of the bags, all the coins weigh 10 grams. However, in one of them, each coin weighs 9.9 grams, i.e. 0.1 gram less than 10 grams. You are given a highly accurate weighing machine that can tell you the weight of an object placed on it, correct to a 100 th of a gram. However, you only have enough money to use it once. How can you find out which bag has the defective coins?
Solution: Label the bags from 1 to 10. And label the coins in the bags with the number of the bag. Take one coin from bag 1, two coins from bag 2 and so on. This will yield 1 + 2 + 3 + … 10 coins = 55 coins. Weigh these 55 coins once on the weighing machine. If all the coins had been 10 grams, the weight you measure would have been 550 grams. But the actual weight will be lower because of the defective coins in one bag. If the weight is 0.1 gram less than 550 grams, the defective bag is bag 1, if the weight is 0.2 grams less than 550 grams, the defective bag is bag 2 etc. In general, if the weight is 0.1Y grams less than 550 grams, the defective bag is bag Y.