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A census taker reads an address on his list and knocks on the door of the house at that address. A father opens the door. The census taker asks the father how many children he has. The father tells the census taker that he has three children and the sum of their ages equals the number on his house. That is not enough information for the census taker. He asks for more information. The father says that the product of the ages of his three children is 36. The census taker thinks that over. He realizes that is still not enough information. So he asks for another clue. The father says "my one oldest child plays the piano." Now the census taker suddenly knows the answer. He writes it down and thanks the father. How did the census taker figure out the ages of the three children? Well, how many combinations are there of three integer ages that equal 36 when multiplied together? There are eight combinations that have a product of 36. Write them down, starting with (1, 1, 36) and (1, 2, 18) and so on up to (3.3,4). Calculate the sum of each of the eight combinations. The sum of the correct combination must equal the house number. If the sum of one and only one combination equaled the house number, the census taker would have his answer. He can se e the house number. But he is still confused and needs another clue. So, there must be two or more combinations that sum up to the house number. That is why he must ask for another clue. The father gives him the last clue: “My one oldest child plays the piano." Look at the only possible combinations that have the same sum. Which one of these combinations would be consistent with "my one oldest child plays the piano?" (Implying that there is only one oldest child). Write down all the combinations, look at the sums, and then solve the riddle. The answer: There are two combinations (1, 6 ,6) and (2, 2, 9) that have the same sum which equals to 13. All the other combinations except the above two have unique (different) sums. If one of the unique sums had equaled the house number, the census taker would have known the answer and he would not have needed the last clue. So, the house number must be 13 and that is why the census taker could not solve the riddle without the last clue. The possible answers are (1, 6 ,6) and (2, 2, 9). Only (2, 2, 9) has "one oldest child."

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