  A fundamental idea in toric topology is that classes of manifolds with
well-behaved torus actions (simply, toric spaces) are classified by pairs of
simplicial complexes and (non-singular) characteristic maps. The authors in
their previous paper provided a new way to find all characteristic maps on a
simplicial complex $K(J)$ obtainable by a sequence of wedgings from $K$. The
main idea was that characteristic maps on $K$ theoretically determine all
possible characteristic maps on a wedge of $K$.
  In this work, we further develop our previous work for classification of
toric spaces. For a star-shaped simplicial sphere $K$ of dimension $n-1$ with
$m$ vertices, the Picard number $\operatorname{Pic}(K)$ of $K$ is $m-n$. We
refer to $K$ a seed if $K$ cannot be obtained by wedgings. First, we show that,
for a fixed positive integer $\ell$, there are at most finitely many seeds of
Picard number $\ell$ supporting characteristic maps. As a corollary, the
conjecture proposed by V. V. Batyrev in 1991 is solved affirmatively.
  Second, we investigate a systematic way to find all characteristic maps on
$K(J)$ using combinatorial objects called (realizable) puzzles that only depend
on a seed $K$. These two facts lead to a practical way to classify the toric
spaces of fixed Picard number.
