A linear threshold for uniqueness of solutions to random jigsaw puzzles

We consider a problem introduced by Mossel and Ross [Shotgun assembly of labeled graphs, arXiv:1504.07682]. Suppose a random n×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 ≤ q ≤ 2 √ e n, all solutions are similar when q ≥ (2+ε)n, and the solution is unique when q = ω(n).