Gourds: A Sliding-Block Puzzle with Turning

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

doi:10.4230/LIPIcs.ISAAC.2020.33

Gourds: A Sliding-Block Puzzle with Turning

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

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

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.

Original language	English
Title of host publication	31ST INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION, ISAAC 2020
Editors	Yixin Cao, Siu-Wing Cheng, Mimming Li
Publisher	Schloss Dagstuhl
Number of pages	16
Volume	181
ISBN (Print)	9783959771733
DOIs	https://doi.org/10.4230/LIPIcs.ISAAC.2020.33
Publication status	Published - 2020
Event	31st International Symposium on Algorithms and Computation - PEOPLES R CHINA, Hong Kong, China Duration: 14 Dec 2020 → 18 Dec 2020 Conference number: 31

Publication series

Series	Leibniz International Proceedings in Informatics (LIPIcs)
Volume	181
ISSN	1868-8969

Conference

Conference	31st International Symposium on Algorithms and Computation
Abbreviated title	ISAAC 2020
Country/Territory	China
City	Hong Kong
Period	14/12/20 → 18/12/20

Keywords

computational complexity
divide-and-conquer
Hamiltonian cycle
puzzle game
(combinatorial) reconfiguration
sliding-block puzzle

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10.4230/LIPIcs.ISAAC.2020.33Licence: CC BY

Cite this

@inproceedings{c0c8d3ed41de4ad690052434c0add146,

title = "Gourds: A Sliding-Block Puzzle with Turning",

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.",

keywords = "computational complexity, divide-and-conquer, Hamiltonian cycle, puzzle game, (combinatorial) reconfiguration, sliding-block puzzle",

author = "Joep Hamersma and \{van Kreveld\}, Marc and Yushi Uno and \{van der Zanden\}, \{Tom C.\}",

year = "2020",

doi = "10.4230/LIPIcs.ISAAC.2020.33",

language = "English",

isbn = "9783959771733",

volume = "181",

series = "Leibniz International Proceedings in Informatics (LIPIcs)",

publisher = "Schloss Dagstuhl",

editor = "Yixin Cao and Siu-Wing Cheng and Mimming Li",

booktitle = "31ST INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION, ISAAC 2020",

address = "Germany",

note = "31st International Symposium on Algorithms and Computation, ISAAC 2020 ; Conference date: 14-12-2020 Through 18-12-2020",

}

Hamersma, J, van Kreveld, M, Uno, Y & van der Zanden, TC 2020, Gourds: A Sliding-Block Puzzle with Turning. in Y Cao, S-W Cheng & M Li (eds), 31ST INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION, ISAAC 2020. vol. 181, 33, Schloss Dagstuhl, Leibniz International Proceedings in Informatics (LIPIcs), vol. 181, 31st International Symposium on Algorithms and Computation, Hong Kong, China, 14/12/20. https://doi.org/10.4230/LIPIcs.ISAAC.2020.33

Gourds: A Sliding-Block Puzzle with Turning. / Hamersma, Joep; van Kreveld, Marc; Uno, Yushi et al.
31ST INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION, ISAAC 2020. ed. / Yixin Cao; Siu-Wing Cheng; Mimming Li. Vol. 181 Schloss Dagstuhl, 2020. 33 (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 181).

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

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T1 - Gourds

T2 - 31st International Symposium on Algorithms and Computation

AU - Hamersma, Joep

AU - van Kreveld, Marc

AU - Uno, Yushi

AU - van der Zanden, Tom C.

N1 - Conference code: 31

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - computational complexity

KW - divide-and-conquer

KW - Hamiltonian cycle

KW - puzzle game

KW - (combinatorial) reconfiguration

KW - sliding-block puzzle

U2 - 10.4230/LIPIcs.ISAAC.2020.33

DO - 10.4230/LIPIcs.ISAAC.2020.33

M3 - Conference article in proceeding

SN - 9783959771733

VL - 181

T3 - Leibniz International Proceedings in Informatics (LIPIcs)

BT - 31ST INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION, ISAAC 2020

A2 - Cao, Yixin

A2 - Cheng, Siu-Wing

A2 - Li, Mimming

PB - Schloss Dagstuhl

Y2 - 14 December 2020 through 18 December 2020

ER -

Gourds: A Sliding-Block Puzzle with Turning

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Keywords

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