An integer linear programming approach to solving the Eternity Puzzle

Abstract

In this article, we develop a new integer linear programming (ILP) approach to solving the Eternity Puzzle, a hard combinatorial tiling problem using 209 polygonal puzzle pieces called polydrafters to exactly cover an almost-regular dodecagon. Previous approaches to solving the Eternity Puzzle used a refined brute-force approach, utilizing backtracking and statistical techniques that were tailored to the particular problem. Our method yields a flexible mathematical model of the tiling problem, expressed in the linear programming (LP) file format, which can be adapted for a wide range of similar puzzle problems of the Eternity type. The tiling problems are solved using a combination of programming in MATLAB, and the commercial high-performance optimization package CPLEX. We also describe the complex geometry of the Eternity Puzzle and use boundary words to represent the tiles and grids of any Eternity-type tiling puzzle. Our ILP approach is illustrated by solving four small puzzles of the Eternity-type, demonstrating that, with sufficient computing power, our approach would, in principle, yield a solution to the full Eternity Puzzle. Additionally, our publicly available set of MATLAB programs ETERNITY (v1.0.0) is available for download on Zenodo.org. These programs can be used to construct the ILP formulations, plot solutions, and be adapted by other researchers to investigate similar puzzle problems.