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Incredible, but solvable (Posted on 2015-10-30) Difficulty: 4 of 5
A puzzle by Princeton mathematician John Horton Conway:

Last night I sat behind two wizards on a bus, and overheard the following:

A: I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.
B: How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?
A: No.
B: Aha! AT LAST I know how old you are!

Rem: Taking in account the fatherhood limitations, this is uniquely solvable.

No Solution Yet Submitted by Ady TZIDON    
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re: @ Charlie- an explanation and a question | Comment 10 of 11 |
(In reply to @ Charlie- an explanation and a question by Ady TZIDON)

What’s   wrong with (48, 13, 5) i.e. the guys are on bus number 13 and A,  aged 48 has 5 children?

The only things B knows are: (1) that knowing the number of children and the product of their ages would not let him deduce the ages, and (2) the number of the bus. At the aha! stage he has not been told the product of the ages, nor has he been told the number of children. His only factual knowledge (as opposed to metaknowledge of what he wouldn't be able to know with extra information) is the bus number.

For bus 13 it could be any of these four cases:

   13   36 3    1  6  6          
   13   36 3    2  2  9          

   13   48 5    1  1  3  4  4          
   13   48 5    1  2  2  2  6    

and they involve two different ages for A, so he wouldn't know which age that is. But the puzzle says he can deduce A's age (the product of the childrens' ages).  That's possible only if the bus's number was 12, as that has only the two cases, both of which have the same age for A (product of childrens' ages).

  Posted by Charlie on 2015-10-31 21:23:38
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