Expert: Abe Mantell - 8/14/2010

Question
A Friend of mine challenged me and some of other friends if we could answer a question.
He said -
I want u guys to give me 4 numbers:
the conditions to be satisfied:
1.the sum of 1st and 2nd must be a square (a perfect square)(like 144=12square)

2.Similarly the sum of 1st and 3rd,1st and 4th,2nd and 3rd,2nd and 4th and 3rd and 4th also should be a square(perfect).

3.Also if u add all the 4 numbers,then that number also should be a square.

I want to ask is there any alternate method to find the answer?
or could u solve this for me?

Answer
So, you want 4 whole numbers so that the sum of any 2 or all 4 of them are perfect squares...yes?

Also, I gather we should exclude zero, as once you find a set of
3 numbers that work, then adding zero to the list will also work.
So, adding zero is not an interesting case.

Anyhow, I wrote a simple program to do a search.  I did the search
for whole numbers between 1 and 25.  The only solution obtained was
1, 8, 8, 8

Abe