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From: skrivan@ssc-vax.UUCP
Newsgroups: net.puzzle
Subject: Sums of Subsets Problem
Message-ID: <157@ssc-vax.UUCP>
Date: Wed, 17-Oct-84 12:19:48 EDT
Article-I.D.: ssc-vax.157
Posted: Wed Oct 17 12:19:48 1984
Date-Received: Sun, 21-Oct-84 13:53:18 EDT
Distribution: net
Organization: Boeing Aerospace Co., Seattle, WA
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The following is a problem I have been carrying along for many years
with little or no progress.  As I reach the prime years of my life
I want to know this problem's secret (from me) solution.  Help!

PROBLEM:  Consider a set N of n positive integers with largest element 
	  X.  There are m = 2**n subsets (empty set included).  Form
	  the m sums of elements of these subsets.

	  What is the set N with smallest X such that the m sums
	  are unique?

The following are some of the known solutions:

		  n                 set
		  _           ___________________     

		  1           {1}
		  2           {1,2}
		  3           {1,2,4} and {2,3,4}
		  4           {3,5,6,7}
		  5           {6,9,11,12,13}

As n increases, candidate sets N can be constructed with unique sums
of subsets.  However, I don't know if they are the solution sets.
Of course, I would prefer a mathematical solution for the general case.

						    Jim Skrivan