Ali the thief had managed to break into the Sultan’s treasure room. He was looking for a bag containing 1250 diamonds, each weighing 0.8 grams. Unfortunately, the Sultan had filled nine other bags with imitation diamonds. There were 1000 in each bag, each one weighing 1 gram. So all 10 bags weighed exactly 1000 grams and all the bags looked exactly alike. Just as Ali was scratching his head, the Sultan burst in with his guards. As he was a merciful Sultan, he gave Ali a chance to save his life. Here is what he said; “If you can find the bag which contains the real diamonds by using these scales just once, you may keep the diamonds. If not, you will be beheaded. You can take stones from any of the bags and put them in other bags if you wish but you can only weigh once!”
How did Ali save his neck?
Clue for the Puzzle:
Suppose there were only three bags. If Ali took one stone from bag no. 1, two from bag no. 2, and three from bag no. 3 and weighed them together, would he be able to tell which the bag of diamonds was?
Answer for the Puzzle:
Ali put the bags in a row and removed one stone from the first bag, two from the second, three from the third, etc. He therefore had 55 stones. If all the stones weighed 1 gram, the total weight would be 55 grams. But Ali knew that some were 0.2 grams lighter. So supposing the total weight had been 54.6 grams, he would have known that it was bag no.2 which contained the diamonds (53 stones at 1 gram and two stones at 0.8 gram each = 1.6 grams). If the total had weighed 54.2 grams, he would have known it was the fourth bag he was looking for (51 stones at 1 gram and 4 stones at 0.8 gram each = 3.2 grams). In this way he could locate the bag with the real diamonds with no trouble at all.