When are eight integers entirely determined by their pairwise sums?
Hint: if the collections $(a_1, \dots, a_k)$ and $(b_1, \dots, b_k)$ have identical pairwise sums, then the collections $(a_1, \dots, a_k, b_1+m, \dots, b_k+m)$ and $(b_1, \dots, b_k, a_1+m, \dots, a_k+m)$ also have identical pairwise sums. (The number $m$ has to be such that $a_i \neq b_j \pm m$ for all $i,j$, so that the numbers in each collection would be different.)