A Global Approach for Solving Edge-Matching Puzzles

We consider apictorial edge-matching puzzles, in which the goal is to arrange a collection of puzzle pieces with colored edges so that the colors match along the edges of adjacent pieces. We devise an algebraic representation for this problem and provide conditions under which it exactly characterizes a puzzle. Using the new representation, we recast the combinatorial, discrete problem of solving puzzles as a global, polynomial system of equations with continuous variables. We further propose new algorithms for generating approximate solutions to the continuous problem by solving a sequence of convex relaxations. 1. Introduction. Jigsaw puzzles [47], dating back to the 1760s, are among the most popular single-player puzzles. The edge-matching puzzle, introduced in the 1890s, is a variation of the jigsaw puzzle in which the goal is to arrange a given collection of tiles with colored edges so that the colors match up along the edges of adjacent tiles. An example of an edge-matching puzzle with 36 square pieces is shown in Figure 1.1. Edge-matching puzzles are challenging compared to standard jigsaw puzzles as only an entire solution guarantees the correctness of any local part of the solution. Despite recent breakthroughs in algorithmic solutions for pictorial puzzles [40, 20], in which one aims to reorganize a scrambled image, relatively little attention had been given to apictorial edge-matching puzzles. This NP-hard problem [17] sparked a lot of interest following the launch of the Eternity puzzle challenges. The challenge posed in the " Eternity I " puzzle was to tile a large dodecagon with 209 irregularly shaped smaller polygonal pieces; it was marketed as being practically unsolvable, but was solved within a year, for a prize of 1 million pounds. For the " Eternity II " puzzle one must correctly place 256 square pieces, whose edges are marked with different patterns, into a 16×16 grid. The puzzle was launched in 2007 and to date remains unsolved. A prize of 2 million dollars was on offer up until December 2010. In this paper we propose a novel representation for apictorial edge-matching puzzle games in terms of algebraic varieties, i.e., as solutions of systems of polynomial equations derived from the pieces of the puzzle. We explain how to generate systems of polynomial equations which are satisfied by puzzle solutions. We refer to systems for which the converse also holds, that is, any solution of the system is a solution of the puzzle, as …