# A linear threshold for uniqueness of solutions to random jigsaw puzzles

## TL;DR

This paper analyzes the conditions under which a random n×n jigsaw puzzle has a unique solution, showing a phase transition depending on the number of piece types q, with a linear threshold for uniqueness.

## Contribution

It establishes a linear threshold for the number of piece types q that guarantees unique solutions in random jigsaw puzzles, extending previous understanding of puzzle solvability.

## Key findings

- Unique solutions occur when q is much larger than n.
- Multiple solutions appear when q is less than approximately 2/√e times n.
- All solutions are similar when q exceeds (2+ε)n.

## Abstract

We consider a problem introduced by Mossel and Ross [Shotgun assembly of labeled graphs, arXiv:1504.07682]. Suppose a random $n\times n$ 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 $2\leq q \leq \frac{2}{\sqrt{e}}n$, all solutions are similar when $q\geq (2+\varepsilon)n$, and the solution is unique when $q=\omega(n)$.