A special case says it all ... Let $ w_1 < w_2 < \ldots < w_{12} $ be an increasing sequence of $12$ integers ("weights") such that the total weight $W=\sum_{k=1}^{12}w_k$ is even.

Say that $I \subseteq \lbrace 1,2, \ldots ,12 \rbrace$ is an exact subset iff the sum $\sum_{k \in I}w_k$ equals $\frac{W}{2}$. My question is : is there a sequence for which $ \lbrace 1,2,5,7,10,12 \rbrace $ is exact and is the only exact subset (up to complementation) ?