A lovely little Question I Like

Well its been a quite time on here, that can only mean it's exam time. As i'm in final year I've been taking it full force - university at 10 am every morning, decent breakfast, the whole 9 yards. the best part is that I feel great!. As I have been revising, that unfortunatly means nothing of interest will occur. Although Next weekend (22$^{nd}$ of January) should be interesting...


In the meantime, here is one of my favourite simple maths questions, that I find really enjoyable.


A Curious Exchange.


Mrs. Smith: The product of their ages is 36, and the sum of their ages is the address on our door here.

Census Taker: (after some figuring) I’m afraid I can’t determine their ages from that …

Mrs. Smith: My eldest daughter has red hair.

Census Taker: Oh, thanks, now I know.

How old are the three girls?

If this were a blog that people actually read and commented on, then I would have probably delayed the answer for a while, however as I am probably the only one to read this, I shall give the answer below. 

I would like to hope that people would at least attempt this though.

Its easier than you think. 

So analysing the question, we are told "The product of their ages is 36"
So we can say that we need 3 numbers that have the product of 36.
1 - 1 - 36
1 - 2 - 18
1 - 3 - 12
1 - 4 - 9
1 - 6 - 6
2 - 2 - 9
2 - 3 - 6


Now we have this list, we then analyse the second part of what we are told. " and the sum of their ages is the address on our door here"
Now we take the sum of all these:

1 - 1 - 36  = 38

1 - 2 - 18 = 21

1 - 3 - 12 = 16

1 - 4 - 9 = 14

1 - 6 - 6 = 13

2 - 2 - 9 = 13

2 - 3 - 6 = 11

We can see that 2 of the groups of numbers sum to 13. We now no that the ages of the 3 daughters are either (1, 6, 6) or (2, 2, 9).
Like the census taker in the question, we cannot yet work out who the ages, however what we are told next makes that possible. "My eldest daughter has red hair"
As a 'Red' herring, we can finally say that the age of the daughters are 2, 2, and 9. This is as we know that there is an eldest daughter, so we can eliminate the other set of numbers.


Isnt it nice?