TITLE: Can you determine a set of values from a set of sums?
QUESTION [7 upvotes]: Consider the following problem: 
There is an vector $A$ (which you will not see) of $n$ positive integers. You are given the set of sums of the (contiguously indexed) subvectors of $A$. For example, say
$$A = (3,2,1,2)$$
The subvectors are $(3),(2),(1),(2), (3,2), (2,1), (1,2),(3,2,1), (2,1,2),(3,2,1,2)$.
We would be given the sums $\{1, 2, 3, 5, 6, 8\}$.  Let us call this set of sums $f(A)$.  
Is it always possible to uniquely determine the set of integers in $A$ from $f(A)$ and $n$?  

The answer turns out to be no. I posted a followup to When does a set of sums uniquely determine a set of values? .

REPLY [6 votes]: The answer is no.
Take the following subvectors: 
$$A = (1, 1, 3)$$$$B = (1,2,2)$$It's easy to see that both vectors have a sum vector of $[1,2,3,4,5]$.