2 people walk on the street and talk:
John: "Did you know my 3 children have a birthday today?"
Paul: "No, I did not"
John: "Do you have any idea how old they are?"
Paul: "No"
John: "If you add up their ages you'll get 13. Do you know now?"
Paul: "No"
John: "And what if I tell you that if you multiply their ages you'll get this number ( points at a certain number )"
Paul: "I still don't know how old they are"
John: "My oldest son is playing the piano. Do you know how old they are now?"
Paul: "Yes I do!"
How old are the 3 children and what number did John pointed at?
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The information is that there are three integers, x , y , and z , such that
1) x + y + z = 1 3
2) the product xyz can also be achieved by a different triple, a + b + c = 1 3 , a b c = x y z
3) the greatest value of the three is old enough for said child to be playing the piano
There are 1 4 triples ( x , y , z ) of positive integers with x ≤ y ≤ z and x + y + z = 1 3 . Those fourteen range from ( 1 , 1 , 1 1 ) to ( 4 , 4 , 5 ) , and have thirteen distinct products, ranging from 1 1 to 8 0 . The only product that is repeated is 3 6 , as 1 + 6 + 6 = 1 3 = 2 + 2 + 9 , and 1 ∗ 6 ∗ 6 = 3 6 = 2 ∗ 2 ∗ 9 .
Thus the children must either be ( 1 , 6 , 6 ) or ( 2 , 2 , 9 ) years old.
Somehow, the final clue is supposed to rule out ( 1 , 6 , 6 ) . I disagree here. Either the implication is that 6-year olds cannot play the piano, or that, somehow, the usage of the phrase "older brother" precludes ( 1 , 6 , 6 ) . My personal background includes both piano lessons at such a young age, and a pair of older siblings that are fraternal twins, one of each gender. It's very easy to imagine one of my parents referring to my older brother as "the older son."
He also took piano lessons.