no sets of six integers with every pair summing to a square
Supposing that there is a set (a,b,c,d,e,f) and a+b, a+c, a+d, b+c... are all squares. Then each member of the set can be paired with 5 others but a+b=b+a, so there are 5*6/2=15 combinations. 5(a+b+c+d+e+f)=sum of all 15 squares.
Re: no sets of six integers with every pair summing to a square
I do not know the answer but there is a set of 4 numbers that I know.
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.