  A sum-network is a directed acyclic network where each source independently
generates one symbol from a given field $\mathbb F$ and each terminal wants to
receive the sum $($over $\mathbb F)$ of the source symbols. For sum-networks
with two sources or two terminals, the solvability is characterized by the
connection condition of each source-terminal pair [3]. A necessary and
sufficient condition for the solvability of the $3$-source $3$-terminal
$(3$s$/3$t$)$ sum-networks was given by Shenvi and Dey [6]. However, the
general case of arbitrary sources/sinks is still open. In this paper, we
investigate the sum-network with three sources and $n$ sinks using a region
decomposition method. A sufficient and necessary condition is established for a
class of $3$s$/n$t sum-networks. As a direct application of this result, a
necessary and sufficient condition of solvability is obtained for the special
case of $3$s$/3$t sum-networks.
