There are four children in a family. The sum of the squares of the ages of the three youngest children equals the square of the oldest child. How old are the children?

Answer 1

the solution is a Pythagorean quadruple ... a^2 + b^2 + c^2 = d^2

there are an infinite number solutions ... but not for children's ages

google the term for a list of suitable solutions

Answer 2

How about ages: 1,2,2,3 or 2,3,6,7, or 1,4,8,9 etc
There are 31 primitive Pythagorean quadruples in which all entries are less than 30.

( 1 , 2 , 2 , 3 ) ( 2 , 10 , 11 , 15 ) ( 4 , 13 , 16 , 21 ) ( 2 , 10 , 25 , 27 )
( 2 , 3 , 6 , 7 ) ( 1 , 12 , 12 , 17 ) ( 8 , 11 , 16 , 21 ) ( 2 , 14 , 23 , 27 )
( 1 , 4 , 8 , 9 ) ( 8 , 9 , 12 , 17 ) ( 3 , 6 , 22 , 23 ) ( 7 , 14 , 22 , 27 )
( 4 , 4 , 7 , 9 ) ( 1 , 6 , 18 , 19 ) ( 3 , 14 , 18 , 23 ) ( 10 , 10 , 23 , 27 )
( 2 , 6 , 9 , 11 ) ( 6 , 6 , 17 , 19 ) ( 6 , 13 , 18 , 23 ) ( 3 , 16 , 24 , 29 )
( 6 , 6 , 7 , 11 ) ( 6 , 10 , 15 , 19 ) ( 9 , 12 , 20 , 25 ) ( 11 , 12 , 24 , 29 )
( 3 , 4 , 12 , 13 ) ( 4 , 5 , 20 , 21 ) ( 12 , 15 , 16 , 25 ) ( 12 , 16 , 21 , 29 )
( 2 , 5 , 14 , 15 ) ( 4 , 8 , 19 , 21 ) ( 2 , 7 , 26 , 27 )