  For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. In
this paper, for a sequence of integers $B=\{1<b_1<b_2<\cdots\}$ and $3b_1+5\leq
b_2\leq 6b_1+10$, we determine the critical value for $b_3$ such that there
exists an infinite sequence $A$ of positive integers for which
$P(A)=\mathbb{N}\setminus B$. This result shows that we partially solve the
problem of Fang and Fang [`On an inverse problem in additive number theory',
Acta Math. Hungar. 158(2019), 36-39].
