This answer deals only with 'regular' solutions.
Let the 3 numbers to be squared be a, b, c. By basic algebra:
a^2+d=b^2
a^2+2d=c^2
From 1
a^2+2(b^2a^2)=c^2
2b^2=c^2+a^2
This will always be true when a=b=c, but assume a and b are different.
a can be any integer we choose, and there will always be a solution:
5a=b, 7a=c, d=(5^21)a^2
29a=b, 41a=c, d=(29^21)a^2
169a=b, 239a=c, d=(169^21)a^2, etc.
Generally, let x be a number such that 2x^21=y^2 has a solution in the positive integers. Then a, b=(ax), c=(ay), d=(x^21)a^2 is a solution:
(ax)^2a^2 = (ay)^2(ax)^2
a^2(x^21) = a^2(y^2x^2)
Clearing the a^2 terms and collecting the others: 2x^21=y^2; the relationship between x and y is independent of a and is fixed.
As I said above, 'regular' solutions, because a might, e.g. be 7, in which case {b,c,d} = {13,17,120}{17,23,240} etc. also represent valid, but not 'regular' solutions.
I'm not sure about the 'UN cargo' clue, though. If it is a clue, that is...
Edited on March 13, 2013, 12:21 pm

Posted by broll
on 20130313 12:11:52 