I have the book in front of me. It is called Mathematical Mind Benders and is by Peter Winkler. On page 22 there's a problem called "Getting the numbers back" that states:

For which positive integers n is it the case that, given the C(n, 2) pairwise sums of n distinct positive integers, you can recover the integers uniquely?

I must say that this is a really good "puzzle" book. The puzzles are refreshingly original and the difficulty level is significantly higher than most books of this type. There are many problems that I would place at the USAMO/IMO level of difficulty. Consider this an unqualified recommendation!

thanks! it seems like an interesting book. i see that they state the problem carefully (although not allowing repeats in the original collection), while avoiding the term "multi-set". yesterday i asked: ) is the usual-set case of any (further) interest? probably not. for example, if A = {0, 1, 2, 4, 5, 6, 7} , then its 2-at-a-time sumset is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} , and the same for the reflected set A' = {0, 1, 2, 3, 5, 6, 7} . the same technique shows that for all n >= 7 , there are distinct usual-sets that have the same 2-at-a-time sum-(usual)-sets. moreover, it is possible to have distinct usual-sets of different cardinalities, with the same 2-at-a-time sum-(usual)-sets. so it seems natural to consider only the multi-set version of the question. mike