How old are the kids?

Logic Level 2

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?

1,5,7 and 35 4,4,5 and 80 2,2,9 and 36 1,6,6 and 36

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2 solutions

Richard Desper
Oct 4, 2019

The information is that there are three integers, x x , y y , and z z , such that

1) x + y + z = 13 x + y + z = 13

2) the product xyz can also be achieved by a different triple, a + b + c = 13 , a b c = x y z a + b + c = 13, abc = xyz

3) the greatest value of the three is old enough for said child to be playing the piano

There are 14 14 triples ( x , y , z ) (x,y,z) of positive integers with x y z x \leq y \leq z and x + y + z = 13 x + y + z =13 . Those fourteen range from ( 1 , 1 , 11 ) (1,1,11) to ( 4 , 4 , 5 ) (4,4,5) , and have thirteen distinct products, ranging from 11 11 to 80 80 . The only product that is repeated is 36 36 , as 1 + 6 + 6 = 13 = 2 + 2 + 9 1 + 6 + 6 = 13 = 2 + 2 + 9 , and 1 6 6 = 36 = 2 2 9 1*6*6 = 36 = 2*2*9 .

Thus the children must either be ( 1 , 6 , 6 ) (1,6,6) or ( 2 , 2 , 9 ) (2,2,9) years old.

Somehow, the final clue is supposed to rule out ( 1 , 6 , 6 ) (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 ) (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.

Gal Dali
Oct 4, 2019

From all the options of three numbers that add up to 13 there are only 2 options that also have the same product-

  • 1,6,6

  • 2,2,9

since Paul doesn't know how old they are after John tells him that the product is a certain number, we know that there is more than one option that adds

up to 13 and also have a product of the same number.

The 2 aformentioned options add up to 13 and their product is 36.

In order to discriminate one option we'll look at the last statement John says:

"My oldest son is playing the piano. Do you know how old they are now?"

From this we understand that there has to be an older brother and therefore option 1,6,6 is no longer valid and the answer is:

2 , 2 , 9 \boxed{2,2,9 }

Why does (1,6,6) preclude the existence of an older brother?

Richard Desper - 1 year, 8 months ago

Good point. I changed it to the oldest instead of the older. btw, I started taking piano lessons at the age of 7😉

Gal Dali - 1 year, 8 months ago

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