there is a problem related to this (for the 2-at-a-time sums) in the problem section of the most recent american mathematical monthly. in the problem, they use multisets for both the original collection and the collection of 2-at-a-time sums. i haven't worked through it yet, but it's a starting point. the reference is: problem 11389, proposed by elizabeth r. chen and jeffrey c. lagarias, american mathematical monthly 115 (october 2008), no. 8, p. 758. mike
In WW Rouse Ball's A short account of the history of mathematics he gives the following "characteristic problem from Diophantus":
Find four numbers, the sum of every arrangement 3 at a time being given; say 22, 24, 27, 20.
What about
Find four numbers, the sum of every arrangement two at a time being given, say 14, 12, 23, 16, 27, 25 ?
I'm interested in what work might have been done on problems like this. I'm particularly interested in the good algorithms for such problems.
Hard, easy, or what?