Shotgun edge assembly of random jigsaw puzzles

Understand

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.