A census taker asks a lady at her front door how many people live in her house and what their ages are. The woman tells him that her three daughters live in the house, that the product of their ages is thirty-six, and that the sum of their ages is the number of the house next door. The census taker goes next door and looks at the number of the house. When he returns he tells the woman the information she gave him is not sufficient, whereupon the woman tells him, "My oldest daughter is sleeping upstairs." The census taker thanks her and promptly figures out the daughters' ages. What are the ages and how does he know?
Ten bags in front of you on the floor are each full of newly-minted 1oz platinum bars. All of the bars look exactly the same, but all the bars in one of the bags are rejects and each bar weighs 1 gram less than a certified bar. You have an extremely accurate scale capable of measuring weight to the gram, but you are only permitted to use the scale once to make only one weighing. Using the scale only once for one weighing, how do you determine which bag has the bad bars?
You come upon 3 chests containing coins. You cannot see inside any of them. Each has a hole in the top and a sign on the front. They are labeled "GOLD," "SILVER," and "GOLD AND SILVER." Each is labeled incorrectly. You are permitted to reach into one box and remove and inspect one coin. You may not peek or feel around inside any chest. Staying within the rules given, how do you label each chest correctly?