## Ages of Three Daughters

My uncle’s in town and just reminded me of a puzzle that he’d asked me over ten years ago. A census worker goes to a house and the person who answers the door tells the census worker that she has three daughters.

“I need to know their ages,” replies the census worker.

“Well, the product of their ages is my age: 36.”

“That’s not enough information.”

“Well, the sum of their ages is the same as the number as the house on the right.”

The census worker checks the house next door and responds, “That’s still not enough information.”

“My eldest daughter is asleep upstairs.”

The census worker thanks the homeowner and leaves. How old are the three daughters?

I love this puzzle because on the face of it, it would appear that there isn’t enough information to figure out the puzzle. If you want to figure out the puzzle yourself, do not read beyond this point.

The first thing we learn about the ages of the daughters is that the product of the ages is 36.

Let’s consider all such possibilities (let’s assume the puzzle maker only allows for integer ages):

1,1,36
1,2,18
1,3,12
1,6,6
1,4,9
2,2,9
2,3,6
3,3,4

As the census worker notes, there are still too many to disambiguate the ages.

The next clue is that the sum of the ages is the same as the number of the house next door. We don’t know the sum, but the census worker does, and we also know that the house number is insufficient to disambiguate the ages. Let’s look at the sums of the factorizations above:

1+1+36 = 38
1+2+18 = 21
1+3+12 = 16
1+6+6 = 13
1+4+9 = 14
2+2+9 =13
2+3+6 = 11
3+3+4 = 10

Note that the only sum that is not unique above is 13, so that is the only possibility in which the second clue would not have been enough for the census worker to disambiguate the ages. This leaves two possibilities:

1,6,6
2,2,9

The final clue helps with that one. If there’s an eldest daughter (let’s assume the puzzle maker created a world in which twins are considered to be the same age and that no two daughters who aren’t twins were born within twelve months of each other), then the only viable solution is 2,2,9.

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