An official inquires about the residents of an apartment from the housekeeper: »How many people live in the apartment?« - »Three« - »How old are they?« - »The product of their ages is 1296« - »I still can't tell how old they are from this« - »Then I will also tell you that the sum of their ages is exactly the same as the house number« - »I still can't tell how old they are after this,« says the official after a longer thought. - »But do you know how old I am?« - »I do« - »Well, they are all younger than me« - »Thank you, now I know how old they are.« - Question: How old are the residents?
Based on all 42 possible three-factor decompositions of 1296, the official responded to the statement that the sum of the three factors is the house number: "I still cannot know their ages," which clearly means that he found at least two three-factor decompositions whose sum is the house number. Indeed, among the three-factor decompositions of 1296, we find only two: \(1 \cdot 18 \cdot 72\) and \(2 \cdot 8 \cdot 81\), in which the sum of the factors is the same, namely 91. (This was therefore the house number.) The caretaker's next response, which states that all the residents are younger than him, combined with the official's response "… now I know how old they are," decided the question in favor of the first decomposition. (The caretaker's age was clearly between 72 and 81.) Therefore, the ages of the residents are: 72, 18, and 1 year.