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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
3 years, 11 months ago

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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 Zi Song Yeoh · 3 years, 11 months 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! Tim Ye · 3 years, 11 months ago

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@Tim Ye 3,3,8. :) This is very well known. Zi Song Yeoh · 3 years, 11 months ago

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@Tim Ye This looks similar to something on the Internet I found. :D Zi Song Yeoh · 3 years, 11 months ago

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@Tim Ye Why must the police station be at 14? Tan Li Xuan · 3 years, 11 months ago

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@Tan Li Xuan 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.:) ;) :) Brilliant Kumar · 3 years, 11 months ago

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@Brilliant Kumar Ok,I understand.Thanks! Tan Li Xuan · 3 years, 11 months ago

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@Brilliant Kumar Yes Zi Song Yeoh · 3 years, 11 months ago

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@Tan Li Xuan Sorry, it should be police station. :) Zi Song Yeoh · 3 years, 11 months ago

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@Tan Li Xuan Because \(8 + 3 + 3 = 14\). Zi Song Yeoh · 3 years, 11 months ago

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This is known as the Census Taker Problem. Zi Song Yeoh · 3 years, 11 months ago

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What does " tall and blue-eyed " have to do with this question ? Tan Li Xuan · 3 years, 11 months ago

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@Tan Li Xuan Nothing, elder is the keyword . Zi Song Yeoh · 3 years, 11 months ago

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@Zi Song Yeoh 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. Harshit Kapur · 3 years, 11 months ago

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@Harshit Kapur See this for the complete solution. Zi Song Yeoh · 3 years, 11 months ago

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@Harshit Kapur Sorry, I misread, I thought it was eldest Zi Song Yeoh · 3 years, 11 months ago

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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!! Shourya Pandey · 3 years, 11 months ago

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@Shourya Pandey 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 Tan Li Xuan · 3 years, 11 months ago

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@Shourya Pandey No Zi Song Yeoh · 3 years, 11 months ago

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This is famous. Answer is \(8,3,3\), and the police office is at \(8 + 3 + 3 = 14\). Zi Song Yeoh · 3 years, 11 months ago

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