Answer: 2, 3, 6 and 7.
Step-by-step explanation:
We have four natural numbers A, B, C and D.
such that all of these numbers are smaller than 10, and:
A < B < C < D.
A, B and C are the ages of the smaller ones, and D is the age of the larger kid.
Now we have the relation
A^2 + B^2 + C^2 = D^2
We can write this as:
D = √(A^2 + B^2 + C^2)
And remember that D must be smaller than 10, then now we can play with the numbers A, B and C in order to find D.
Suppose for example:
A = 1, B = 2, C = 3
D = √( 1^2 + 2^2 + 3^2) = √(1 + 4 + 9) = √14
This is not a whole number.
Then we have that A^2 + B^2 + C^2 must be a perfect square.
The perfect squares we can aim for are:
4*4 = 16
5*5 = 25
6*6 = 36
7*7 = 49
8*8 = 64
9*9 = 81
now, looking at that comes to my mind to try to reach the 49.
because we can take the 36, add 4 to that (4 = 2*2) and then add 9 (9 = 3*3)
A^2 + B^2 + C^2 = 49
where we can use:
A = 2. B = 3 and C = 6
2^2 + 3^2 + 6^2 = 4 + 9 + 36 = 49
Then we have:
D = √(2^2 + 3^2 + 6^2 ) = √49 = 7
Then the ages of the children (for smallest to largest) are:
2, 3, 6 and 7.
