  We consider a problem introduced by Mossel and Ross [Shotgun assembly of
labeled graphs, arXiv:1504.07682]. Suppose a random $n\times n$ jigsaw puzzle
is constructed by independently and uniformly choosing the shape of each "jig"
from $q$ possibilities. We are given the shuffled pieces. Then, depending on
$q$, what is the probability that we can reassemble the puzzle uniquely? We say
that two solutions of a puzzle are similar if they only differ by permutation
of duplicate pieces, and rotation of rotationally symmetric pieces. In this
paper, we show that, with high probability, such a puzzle has at least two
non-similar solutions when $2\leq q \leq \frac{2}{\sqrt{e}}n$, all solutions
are similar when $q\geq (2+\varepsilon)n$, and the solution is unique when
$q=\omega(n)$.
