  In recent work by Mossel and Ross, it was asked how large $q$ has to be for a
random jigsaw puzzle with $q$ different shapes of "jigs" to have exactly one
solution. The jigs are assumed symmetric in the sense that two jigs of the same
type always fit together. They showed that for $q=o(n^{2/3})$ there are a.a.s.
multiple solutions, and for $q=\omega(n^2)$ there is a.a.s. exactly one. The
latter bound has since been improved to $q\geq n^{1+\varepsilon}$ independently
by Nenadov, Pfister and Steger, and by Bordernave, Feige and Mossel. Both
groups further remark that for $q=o(n)$ there are a.a.s. duplicate pieces in
the puzzle. In this paper, we show that such puzzle a.a.s. has multiple
solutions whenever $q\leq \frac{2}{\sqrt{e}}\,n - \omega(\log_2 n)$, even if
permuting identical pieces is not considered changing the solution. We further
give some remarks about the number of solutions, and the probability of a
unique solution in this regime.
