R. Guy, *Unsolved problems in number theory*, 3rd edition, Springer, 2004.


In this book, on page 167-168, Problem C5, *Sums determining members of a set*, discusses a question Leo Moser asked: suppose $X\subset\mathbb{Z}$ is an $n$-element set and $A$ be set of all $k$-element **sums** of subsets of $X$. Is there any other $Y\neq X$ such that $Y\subset\mathbb{Z}$ such that its $k$-element sums are exactly $A$?


Only small cases were known: $k=2,3,4,5$ and any $n$.



> 
> **Question.** What is the general status of this problem for $k>5$? Max Alekseyev provided some references.
> 
> 
> **Update.** I'm still interested in the question: are there (non-cosmetic) equivalent formulations of the above problem?
> 
> 
> 



what is the status of this problem? an equivalent formulation?
In a recent paper [A set of 12 numbers is not determined by its set of 4-sums](http://mi.mathnet.ru/eng/znsl6308) (in Russian), Isomurodov and Kokhas identify an error in the argument of Ewell (1968) and present two distinct sets of 12 integers with the same multiset of 4-sums.


They also give a nice overview of the area and provide [useful references](http://www.mathnet.ru/php/getRefFromDB.phtml?jrnid=znsl&paperid=6308&output=htm&option_lang=eng). For the general case $(n,k)$, they mention the following results:


* Selfridge and Straus (1958) describe a union of Diophantine equations such that for $(n,k)$ not satisfying any of the equations (a typical situation), the set $X$ is unique. For $(n,k)$ satisfying at least one of the equations, the uniqueness of $X$ remains an open question (the case $(12,4)$ was such).
* Fraenkel, Gordon, and Straus (1962) prove that for each $k$, there exists only a finite number of values of $n$, for which $X$ may be not unique.


