SOLUTION: There are two families (The Anderson family and the Billings family) that both consist of a Husband, Wife, Son, and Daughter. In both cases, the families ages add up to 100 years.

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Question 939595: There are two families (The Anderson family and the Billings family) that both consist of a Husband, Wife, Son, and Daughter. In both cases, the families ages add up to 100 years. Also in both cases, the squared ages of the wife, son and daughter add together to equal the squared age of the father. The only difference is that in the Anderson family, the daughter is one year younger than the son, and in the Billings family the daughter is two years younger than the son.
How old are all the members of both families?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
In the Anderson family:
%22husband%27s%22 age=highlight%2839%29 ,
%22wife%27s%22 age=highlight%2834%29 ,
%22son%27s%22 age=highlight%2814%29 , and
%22daughter%27s%22 age=highlight%2813%29 .

In the Billings family:
%22husband%27s%22 age=highlight%2842%29 ,
%22wife%27s%22 age=highlight%2840%29 ,
%22son%27s%22 age=highlight%2810%29 , and
%22daughter%27s%22 age=highlight%288%29 .

How did I find that?
I cheated and used a spreadsheet program (Microsoft Excel) to tabulate possibilities.
I started from
h=%22husband%27s%22 age ,
w=%22wife%27s%22 age ,
s=%22son%27s%22 age , and
d=s-2=%22daughter%27s%22 age for the Billings family,
while d=s-1=%22daughter%27s%22 age for the Anderson family.
I made a table for each family with d as one column,
the corresponding s as a second column,
and several columns with various values of w as a title at the top,
and the corresponding results for h=sqrt%28d%5E2%2Bs%5E2%2Bw%5E2%29 below.
When I found an integer for h,
I calculated h%2Bw%2Bd%2Bs to see if it was 100 .
The possibilities can be narrowed, but I could not narrow them too much.
We also know that the husband must be the oldest one,
because h%5E2=w%5E2%2Bs%5E2%2Bd%5E2%3Ew%5E2--->h%3Ew .
Also, the children must be younger than the parents, so we expect them to be under 25.
I used d values from 1 to 25, and w values from 18 to 40 for my tables.

Maybe some horrible algebra would show an easier path to the solution.
There must be ways to narrow the possibilities further,
and maybe there is a table of Pythagorean quadruples that would help.
Maybe asking the question in some low level artofproblemsolving website forum would prompt a 13 year old to give a succinct brilliant solution that does not require tabulations.
All I could think of is that for the Billings family
we know that h%2Bw%2Bs%2Bs-2=100<-->h%2Bw%2B2s=102<-->h%2Bw=2%2851-s%29 ,
which tells you that the ages of husband and wife are either both even, or both odd.
On the other hand, using a similar reasoning, in the Anderson family,
if the husband's age is even, the wife's age is odd and viceversa.