Gourds: a sliding-block puzzle with turning

TL;DR

Gourds is a novel hexagonal sliding puzzle involving sliding, turning, and pivoting of 1x2 pieces, extending the classic 15-puzzle with rotational moves, and its solvability and complexity are thoroughly analyzed.

Contribution

Introduces the Gourds puzzle, characterizes solvable cases, and analyzes the computational complexity of piece placement with colored constraints.

Findings

Puzzle can be solved in certain configurations

Determining solvability is NP-complete with many colors

Placement problem is polynomial-time solvable with fixed colors

Abstract

We propose a new kind of sliding-block puzzle, called Gourds, where the objective is to rearrange 1 x 2 pieces on a hexagonal grid board of 2n + 1 cells with n pieces, using sliding, turning and pivoting moves. This puzzle has a single empty cell on a board and forms a natural extension of the 15-puzzle to include rotational moves. We analyze the puzzle and completely characterize the cases when the puzzle can always be solved. We also study the complexity of determining whether a given set of colored pieces can be placed on a colored hexagonal grid board with matching colors. We show this problem is NP-complete for arbitrarily many colors, but solvable in randomized polynomial time if the number of colors is a fixed constant.