Answer to Puzzle #34: Man With 3 Daughters of Different Ages

34. A man has three daughters. A second, intelligent man, asked him the ages of his daughters. The first man told him that the product of their ages (them all multiplied together,) was 36. After thinking the second man was unable to find the answer and asked for another clue. The first man replies the sum of their ages is equal to his house door number. Still the second man was unable to answer and asked for another clue. The first man told him that his youngest daughter had blue eyes, and suddenly second man gave the correct answer. Explain how?

This puzzle was emailed to me through this website. I set about replying that I didn't have the answer but these are some things to think about. By the time I'd done that I actually had the answer...

The temptation is to try and solve the puzzle as the second man does. We can't do that, but we have the additional information we glean from the second man.

The first step is to work out the ages that have a product of 36. I hope these are them all (thanks for the emails):

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

There is nothing at this stage either we or the second man can tell, so we apply the second clue and work out the totals:

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

Here in is the crux of the puzzle, we obviously don't know the man's door number. We can assume that the second man does. At this stage we are only assuming, but the fact that he is able later, to definitively answer the puzzle will mean that this is no longer an assumption. And remember we are trying to explain what happened. Even with the door number the second man does not know the ages of the daughters, so it must still be ambiguous, there are ony two combinations that have the same total.

13: 1, 6, 6
13: 2, 2, 9

The next thing revealed by the first man is that his youngest daughter has blue eyes. Again we are not directly inferring anything from this as much as guessing what the second man did. My initial thought was that babies have blue eyes, but then so do adults, this tells us nothing. It's reasonable to assume that the second man took this to mean that the two youngest daughters were not the same age, that there was a youngest daughter. This is not the case with the ages 2, 2 & 9. The only possible solution is 1, 6 & 6

Assumptions

It can't be overstated the importance of the fact that the second man gave the correct answer. We are explaining how he arrived at it. The only way, for example, that we know he knew the door number is because he was able to give the correct answer.

Of course it is possible for a woman to have 2 children within 12 months either twins or not. There is an extent to which as a number puzzle, the puzzle is assuming integer ages. Again we can only get away with this assumption because we are not solving the puzzle itself, so much as piecing together how the second man solved it.

Alternate Version

I found another version of this:

A census taker called at a man's house and said, "What are the ages of your 3 daughters?"
The man said "If you multiply their ages together it equals 72 and if you add them it equals your door number". The census taker said "Well if you cannot give me further information I still don't know".
The man said "Well my eldest daughter has a dog with a wooden leg".
The census taker said, "I know now".

What were their ages?

They both got this wrong in a very similar way. It's basic maths being able to factorise 36, which I suspect they could both do in a different context. Just not here. Weird.

If you're curious what Bard made of this puzzle...

If you're curious what ChatGPT made of this puzzle...

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