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?

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

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 )

Let's assume the ages of the four children as follows:

Let the ages of the three youngest children be represented by a, b, and c.
Let the age of the oldest child be represented by d.

According to the given information, the sum of the squares of the ages of the three youngest children equals the square of the oldest child.

So, we can write the equation as follows:

a^2 + b^2 + c^2 = d^2

Since there are no other constraints or details provided, we can explore possible solutions by finding integer values that satisfy this equation.

Let's take a systematic approach to find the possible values for a, b, c, and d:

1. Start by assuming random values for a, b, and c (e.g. 1, 2, 3).
2. Calculate the squares of these assumed values.

a^2 = 1^2 = 1
b^2 = 2^2 = 4
c^2 = 3^2 = 9

3. Add the squares of a, b, and c to calculate the sum.

Sum = 1 + 4 + 9 = 14

4. Find the square root of the calculated sum to get the value of d.

d = √14 ≈ 3.74

Based on these assumptions, the values of a, b, c, and d do not satisfy the given equation because d is not an integer.

To explore possible solutions further, we can try different values for a, b, and c until we find a combination that satisfies the equation.

To solve this problem, we can start by assigning variables to represent the ages of the four children. Let's use a, b, c, and d to represent the ages of the oldest, second oldest, second youngest, and youngest child, respectively.

According to the problem, the sum of the squares of the ages of the three youngest children equals the square of the oldest child. Mathematically, this can be represented as:

a^2 = b^2 + c^2 + d^2

Now, we can try different combinations of ages for the four children and see which one satisfies the equation. We need to make sure that b, c, and d are all different ages, as the problem states that there are four different children in the family.

Let's start by assigning different ages to the four children:
a = 6, b = 4, c = 3, d = 1

Now, let's check if these ages satisfy the equation:
6^2 = 4^2 + 3^2 + 1^2
36 = 16 + 9 + 1
36 = 26

As we can see, the equation does not hold true for this combination of ages. We need to try a different combination.

Let's try another set of ages:
a = 10, b = 8, c = 6, d = 2

Now, let's check if these ages satisfy the equation:
10^2 = 8^2 + 6^2 + 2^2
100 = 64 + 36 + 4
100 = 100

In this case, the equation holds true. Therefore, the ages of the four children are 10, 8, 6, and 2.