Even 1×n Edge-Matching and Jigsaw Puzzles are Really Hard

Jigsaw puzzles [9] and edge-matching puzzles [5] are two ancient types of puzzle, going back to the 1760s and 1890s, respectively. Jigsaw puzzles involve fitting together a given set of pieces (usually via translation and rotation) into a desired shape (usually a rectangle), often revealing a known image or pattern. The pieces are typically squares with a pocket cut out of or a tab attached to each side, except for boundary pieces which have one flat side and corner pieces which have two flat sides. Most jigsaw puzzles have unique tab/pocket pairs that fit together, but we consider the generalization to “ambiguous mates” where multiple tabs and pockets have the same shape and are thus compatible. Edge-matching puzzles are similar to jigsaw puzzles: they too feature square tiles, but instead of pockets or tabs, each edge has a color or pattern. In signed edge-matching puzzles, the edge labels come in complementary pairs (e.g., the head and tail halves of a colored lizard), and adjacent tiles must have complementary edge labels on their shared edge (e.g., forming an entire lizard of one color). This puzzle type is essentially identical to jigsaw puzzles, where complementary pairs of edge labels act as identically shaped tab/pocket pairs. In unsigned edge-matching puzzles, edge labels are arbitrary, and the requirement is that adjacent tiles must have identical edge labels. In both cases, the goal is to place (via translation and rotation) the tiles into a target shape, typically a rectangle. A recent popular (unsigned) edge-matching puzzle is Eternity II [8], which featured a US$2,000,000 prize for the first solver (before 2011). The puzzle remains unsolved (except presumably by its creator, Christopher Monckton). The best partial solution to date [7] either places 247 out of the 256 pieces without error, or places all 256 pieces while correctly matching 467 out of 480 edges.