# A question and a complicated answer

John asked his neighbour Peter, how old are his three sons. He said that the product of their ages is 72, and the sum is equal to the number where the Police Office is. John makes some calculation in mind and he said that he need further information to say the solution. Peter said: My elder son is tall and blue-eyed. How old are they? And where I can find the Police Office?

Note by Vali Dobre
7 years ago

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I created a list of the ages/sums:

• 1 1 72| 74
• 1 2 36| 39
• 1 3 24| 28
• 1 4 18| 23
• 1 6 12| 18
• 2 2 18| 22
• 2 3 12| 17
• 2 4 9 | 15
• 2 6 6 | 14 **
• 3 3 8 | 14 **
• 3 4 6 | 13

Notice that Peter said he needed further information. This is because the sons could either be 2, 6, and 6, or 3, 3, and 8, however it was ambiguous as to which. When John says the elder son is tall, Peter is able to confirm there is just one "elder" son and thus, it must be the {3, 8, 8} combination.

Great problem!

- 7 years ago

3,3,8. :) This is very well known.

- 7 years ago

Why must the police station be at 14?

- 7 years ago

It ain't that where should be the police station. Instead it says that sum is something known to John and yet he asks for another clue. So, there must be two values of ages with same sums as John needs more information. So, police station has to be at 14 and so, we can proceed to next condition that elder one has something. So, we come to know that there is one and only one elder son. So, it must be 3,3,8.:) ;) :)

- 7 years ago

Yes

- 7 years ago

Ok,I understand.Thanks!

- 7 years ago

Because $8 + 3 + 3 = 14$.

- 7 years ago

Sorry, it should be police station. :)

- 7 years ago

This looks similar to something on the Internet I found. :D

- 7 years ago

Did he forget to include the assumption that their ages are positive integers (i personally think it is important) or isn't it important.

E.g. Something like 2 years 7 months, 7 years 3 days 4 hours 26 minutes and 9 seconds if he didn't include that

- 7 years ago

What does " tall and blue-eyed " have to do with this question ?

- 7 years ago

Nothing, elder is the keyword .

- 7 years ago

the word is elder - which suggests, that two of them are of same age, atleast to me. Though why they can't be 2, 2, and 18, i do not know.

- 7 years ago

Sorry, I misread, I thought it was eldest

- 7 years ago

See this for the complete solution.

- 7 years ago

This is known as the Census Taker Problem.

- 7 years ago

why only 3,3,8? and not 2,2,18? is it because 18 is an adult ? but that does not mean not remaining a son, does it!!

- 7 years ago

Because John said he needed more information,that means the answer has at least 2 solutions which in this case is 2,6,6 and 3,3,8.So when Peter said his elder son was tall and blue-eyed,it means there is only 1 elder son,hence 3,3,8

- 7 years ago

No

- 7 years ago

This is famous. Answer is $8,3,3$, and the police office is at $8 + 3 + 3 = 14$.

- 7 years ago