Unique reconstruction threshold for random jigsaw puzzles

A random jigsaw puzzle is constructed by arranging n2 square pieces into an n× n grid and assigning to each edge of a piece one of q available colours uniformly at random, with the restriction that touching edges receive the same colour. We show that if q = o(n) then with high probability such a puzzle does not have a unique solution, while if q ≥ n1+ε for any constant ε > 0 then the solution is unique. This solves a conjecture of Mossel and Ross (Shotgun assembly of labeled graphs, arXiv:1504.07682).