Shotgun edge assembly of random jigsaw puzzles

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.