Gourds: A Sliding-Block Puzzle with Turning

Authors Joep Hamersma, Marc van Kreveld, Yushi Uno, Tom C. van der Zanden

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Author Details

Joep Hamersma
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Marc van Kreveld
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Yushi Uno
  • Graduate School of Engineering, Osaka Prefecture University, Japan
Tom C. van der Zanden
  • Department of Data Analytics and Digitalisation, Maastricht University, The Netherlands


The original idea for Gourds was formed after an invited talk by Ryuhei Uehara at ICALP 2015.

Cite AsGet BibTex

Joep Hamersma, Marc van Kreveld, Yushi Uno, and Tom C. van der Zanden. Gourds: A Sliding-Block Puzzle with Turning. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We propose a new kind of sliding-block puzzle, called Gourds, where the objective is to rearrange 1×2 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • computational complexity
  • divide-and-conquer
  • Hamiltonian cycle
  • puzzle game
  • (combinatorial) reconfiguration
  • sliding-block puzzle


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