Counting Connected and Disconnected Ways to Assemble a Jigsaw Puzzle

Abstract

A jigsaw puzzle may be assembled in many different ways. Some assembly sequences remain connected throughout, while others temporarily build separate parts of the puzzle before joining them together. By representing the puzzle as a graph, these assembly sequences become vertex orderings that can be counted using recent results from graph theory. We apply this framework to enumerate three natural assembly strategies: connected assembly, assembly beginning from several disconnected pieces, and assembly in which new disconnected sections may be started during the process. The resulting counts reveal that, even for modest puzzle sizes, connectivity-preserving assembly sequences are greatly outnumbered by those that pass through disconnected intermediate stages.