Interestingly enough, the 2-at-a-time sum problem for n=4 isn't necessarily unique! For example, {1, 2, 3, 6} and {0, 3, 4, 5} produce the same sum set, {3, 4, 5, 7, 8, 9}, when the sums are taken 2 at a time. I have a book (at home, alas) that talks about this problem. Basically, for n a power of 2, the numbers are not necessarily determined by the sums taken two at a time. If n is not a power of 2, then the numbers are uniquely determined. I can dig up the reference later, if anyone is interested. Dave On Tue, Oct 14, 2008 at 8:59 AM, Michael Reid <reid@gauss.math.ucf.edu>wrote:

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?

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