Document

**Published in:** LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Various forms of graph coloring problems have been studied over the years in the society of graph theory. Recently, some original puzzles are popularized in Japanese 100-yen shops, and one of them can be formalized as a graph coloring problem in a natural way. However, this natural graph coloring problem has not been investigated in the context of the graph theory. In this paper, we investigate this puzzle as a graph coloring problem. We first prove that this graph coloring problem is NP-complete even when the graph is restricted to a path or a spider. In these cases, diameter of the graphs seems to play an important role for its difficulty. We then show that the problem can be solved in polynomial time when the graph is restricted to some graph classes of constant diameter.

Yuki Iburi and Ryuhei Uehara. Computational Complexity of Matching Match Puzzle. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 17:1-17:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{iburi_et_al:LIPIcs.FUN.2024.17, author = {Iburi, Yuki and Uehara, Ryuhei}, title = {{Computational Complexity of Matching Match Puzzle}}, booktitle = {12th International Conference on Fun with Algorithms (FUN 2024)}, pages = {17:1--17:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-314-0}, ISSN = {1868-8969}, year = {2024}, volume = {291}, editor = {Broder, Andrei Z. and Tamir, Tami}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.17}, URN = {urn:nbn:de:0030-drops-199251}, doi = {10.4230/LIPIcs.FUN.2024.17}, annote = {Keywords: Graph coloring, Matching Match puzzle, NP-complete, polynomial-time solvable} }

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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Edge-connected configurations of square modules, which can reconfigure through so-called sliding moves, are a well-established theoretical model for modular robots in two dimensions. Dumitrescu and Pach [Graphs and Combinatorics, 2006] proved that it is always possible to reconfigure one edge-connected configuration of n squares into any other using at most O(n²) sliding moves, while keeping the configuration connected at all times.
For certain pairs of configurations, reconfiguration may require Ω(n²) sliding moves. However, significantly fewer moves may be sufficient. We prove that it is NP-hard to minimize the number of sliding moves for a given pair of edge-connected configurations. On the positive side we present Gather&Compact, an input-sensitive in-place algorithm that requires only O( ̄P n) sliding moves to transform one configuration into the other, where ̄P is the maximum perimeter of the two bounding boxes. The squares move within the bounding boxes only, with the exception of at most one square at a time which may move through the positions adjacent to the bounding boxes. The O( ̄P n) bound never exceeds O(n²), and is optimal (up to constant factors) among all bounds parameterized by just n and ̄P. Our algorithm is built on the basic principle that well-connected components of modular robots can be transformed efficiently. Hence we iteratively increase the connectivity within a configuration, to finally arrive at a single solid xy-monotone component.
We implemented Gather&Compact and compared it experimentally to the in-place modification by Moreno and Sacristán [EuroCG 2020] of the Dumitrescu and Pach algorithm (MSDP). Our experiments show that Gather&Compact consistently outperforms MSDP by a significant margin, on all types of square configurations.

Hugo A. Akitaya, Erik D. Demaine, Matias Korman, Irina Kostitsyna, Irene Parada, Willem Sonke, Bettina Speckmann, Ryuhei Uehara, and Jules Wulms. Compacting Squares: Input-Sensitive In-Place Reconfiguration of Sliding Squares. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{a.akitaya_et_al:LIPIcs.SWAT.2022.4, author = {A. Akitaya, Hugo and Demaine, Erik D. and Korman, Matias and Kostitsyna, Irina and Parada, Irene and Sonke, Willem and Speckmann, Bettina and Uehara, Ryuhei and Wulms, Jules}, title = {{Compacting Squares: Input-Sensitive In-Place Reconfiguration of Sliding Squares}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {4:1--4:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.4}, URN = {urn:nbn:de:0030-drops-161644}, doi = {10.4230/LIPIcs.SWAT.2022.4}, annote = {Keywords: Sliding cubes, Reconfiguration, Modular robots, NP-hardness} }

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**Published in:** LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

We study the space reachable by rolling a 3D convex polyhedron on a 2D periodic tessellation in the xy-plane, where at every step a face of the polyhedron must coincide exactly with a tile of the tessellation it rests upon, and the polyhedron rotates around one of the incident edges of that face until the neighboring face hits the xy plane. If the whole plane can be reached by a sequence of such rolls, we call the polyhedron a plane roller for the given tessellation. We further classify polyhedra that reach a constant fraction of the plane, an infinite area but vanishing fraction of the plane, or a bounded area as hollow-plane rollers, band rollers, and bounded rollers respectively. We present a polynomial-time algorithm to determine the set of tiles in a given periodic tessellation reachable by a given polyhedron from a given starting position, which in particular determines the roller type of the polyhedron and tessellation. Using this algorithm, we compute the reachability for every regular-faced convex polyhedron on every regular-tiled (≤ 4)-uniform tessellation.

Akira Baes, Erik D. Demaine, Martin L. Demaine, Elizabeth Hartung, Stefan Langerman, Joseph O'Rourke, Ryuhei Uehara, Yushi Uno, and Aaron Williams. Rolling Polyhedra on Tessellations. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{baes_et_al:LIPIcs.FUN.2022.6, author = {Baes, Akira and Demaine, Erik D. and Demaine, Martin L. and Hartung, Elizabeth and Langerman, Stefan and O'Rourke, Joseph and Uehara, Ryuhei and Uno, Yushi and Williams, Aaron}, title = {{Rolling Polyhedra on Tessellations}}, booktitle = {11th International Conference on Fun with Algorithms (FUN 2022)}, pages = {6:1--6:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-232-7}, ISSN = {1868-8969}, year = {2022}, volume = {226}, editor = {Fraigniaud, Pierre and Uno, Yushi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.6}, URN = {urn:nbn:de:0030-drops-159761}, doi = {10.4230/LIPIcs.FUN.2022.6}, annote = {Keywords: polyhedra, tilings} }

Document

**Published in:** LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

Various forms of sorting problems have been studied over the years. Recently, two kinds of sorting puzzle apps are popularized. In these puzzles, we are given a set of bins filled with colored units, balls or water, and some empty bins. These puzzles allow us to move colored units from a bin to another when the colors involved match in some way or the target bin is empty. The goal of these puzzles is to sort all the color units in order. We investigate computational complexities of these puzzles. We first show that these two puzzles are essentially the same from the viewpoint of solvability. That is, an instance is sortable by ball-moves if and only if it is sortable by water-moves. We also show that every yes-instance has a solution of polynomial length, which implies that these puzzles belong to NP . We then show that these puzzles are NP-complete. For some special cases, we give polynomial-time algorithms. We finally consider the number of empty bins sufficient for making all instances solvable and give non-trivial upper and lower bounds in terms of the number of filled bins and the capacity of bins.

Takehiro Ito, Jun Kawahara, Shin-ichi Minato, Yota Otachi, Toshiki Saitoh, Akira Suzuki, Ryuhei Uehara, Takeaki Uno, Katsuhisa Yamanaka, and Ryo Yoshinaka. Sorting Balls and Water: Equivalence and Computational Complexity. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ito_et_al:LIPIcs.FUN.2022.16, author = {Ito, Takehiro and Kawahara, Jun and Minato, Shin-ichi and Otachi, Yota and Saitoh, Toshiki and Suzuki, Akira and Uehara, Ryuhei and Uno, Takeaki and Yamanaka, Katsuhisa and Yoshinaka, Ryo}, title = {{Sorting Balls and Water: Equivalence and Computational Complexity}}, booktitle = {11th International Conference on Fun with Algorithms (FUN 2022)}, pages = {16:1--16:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-232-7}, ISSN = {1868-8969}, year = {2022}, volume = {226}, editor = {Fraigniaud, Pierre and Uno, Yushi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.16}, URN = {urn:nbn:de:0030-drops-159867}, doi = {10.4230/LIPIcs.FUN.2022.16}, annote = {Keywords: Ball sort puzzle, recreational mathematics, sorting pairs in bins, water sort puzzle} }

Document

**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

A swarm of anonymous oblivious mobile robots, operating in deterministic Look-Compute-Move cycles, is confined within a circular track. All robots agree on the clockwise direction (chirality), they are activated by an adversarial semi-synchronous scheduler (SSYNCH), and an active robot always reaches the destination point it computes (rigidity). Robots have limited visibility: each robot can see only the points on the circle that have an angular distance strictly smaller than a constant ϑ from the robot’s current location, where 0 < ϑ ≤ π (angles are expressed in radians).
We study the Gathering problem for such a swarm of robots: that is, all robots are initially in distinct locations on the circle, and their task is to reach the same point on the circle in a finite number of turns, regardless of the way they are activated by the scheduler. Note that, due to the anonymity of the robots, this task is impossible if the initial configuration is rotationally symmetric; hence, we have to make the assumption that the initial configuration be rotationally asymmetric.
We prove that, if ϑ = π (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size. By contrast, we also prove that, if ϑ ≤ π/2, no distributed algorithm solves the Gathering problem, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected.
The latter impossibility result relies on a probabilistic technique based on random perturbations, which is novel in the context of anonymous mobile robots. Such a technique is of independent interest, and immediately applies to other Pattern-Formation problems.

Giuseppe A. Di Luna, Ryuhei Uehara, Giovanni Viglietta, and Yukiko Yamauchi. Gathering on a Circle with Limited Visibility by Anonymous Oblivious Robots. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{diluna_et_al:LIPIcs.DISC.2020.12, author = {Di Luna, Giuseppe A. and Uehara, Ryuhei and Viglietta, Giovanni and Yamauchi, Yukiko}, title = {{Gathering on a Circle with Limited Visibility by Anonymous Oblivious Robots}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.12}, URN = {urn:nbn:de:0030-drops-130907}, doi = {10.4230/LIPIcs.DISC.2020.12}, annote = {Keywords: Mobile robots, Gathering, limited visibility, circle} }

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Complete Volume

**Published in:** LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

LIPIcs, Volume 157, FUN 2021, Complete Volume

10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 1-416, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@Proceedings{farachcolton_et_al:LIPIcs.FUN.2021, title = {{LIPIcs, Volume 157, FUN 2021, Complete Volume}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {1--416}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-145-0}, ISSN = {1868-8969}, year = {2020}, volume = {157}, editor = {Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021}, URN = {urn:nbn:de:0030-drops-127602}, doi = {10.4230/LIPIcs.FUN.2021}, annote = {Keywords: LIPIcs, Volume 157, FUN 2021, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

Front Matter, Table of Contents, Preface, Conference Organization

10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{farachcolton_et_al:LIPIcs.FUN.2021.0, author = {Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {0:i--0:xvi}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-145-0}, ISSN = {1868-8969}, year = {2020}, volume = {157}, editor = {Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.0}, URN = {urn:nbn:de:0030-drops-127613}, doi = {10.4230/LIPIcs.FUN.2021.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

In this paper, we investigate the computational complexity of lattice puzzle, which is one of the traditional puzzles. A lattice puzzle consists of 2n plates with some slits, and the goal of this puzzle is to assemble them to form a lattice of size n x n. It has a long history in the puzzle society; however, there is no known research from the viewpoint of theoretical computer science. This puzzle has some natural variants, and they characterize representative computational complexity classes in the class NP. Especially, one of the natural variants gives a characterization of the graph isomorphism problem. That is, the variant is GI-complete in general. As far as the authors know, this is the first non-trivial GI-complete problem characterized by a classic puzzle. Like the sliding block puzzles, this simple puzzle can be used to characterize several representative computational complexity classes. That is, it gives us new insight of these computational complexity classes.

Yasuaki Kobayashi, Koki Suetsugu, Hideki Tsuiki, and Ryuhei Uehara. On the Complexity of Lattice Puzzles. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 32:1-32:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kobayashi_et_al:LIPIcs.ISAAC.2019.32, author = {Kobayashi, Yasuaki and Suetsugu, Koki and Tsuiki, Hideki and Uehara, Ryuhei}, title = {{On the Complexity of Lattice Puzzles}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {32:1--32:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.32}, URN = {urn:nbn:de:0030-drops-115287}, doi = {10.4230/LIPIcs.ISAAC.2019.32}, annote = {Keywords: Lattice puzzle, NP-completeness, GI-completeness, FPT algorithm} }

Document

**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Given two independent sets I and J of a graph G, imagine that a token (coin) is placed on each vertex in I. Then, the Sliding Token problem asks if one could transforms I to J using a sequence of elementary steps, where each step requires sliding a token from one vertex to one of its neighbors, such that the resulting set of vertices where tokens are placed still remains independent. In this paper, we describe a polynomial-time algorithm for solving Sliding Token in case the graph G is a cactus. Our algorithm is designed based on two observations. First, all structures that forbid the existence of a sequence of token slidings between I and J, if exist, can be found in polynomial time. A no-instance may be easily deduced using this characterization. Second, without such forbidden structures, a sequence of token slidings between I and J does exist.

Duc A. Hoang and Ryuhei Uehara. Sliding Tokens on a Cactus. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 37:1-37:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hoang_et_al:LIPIcs.ISAAC.2016.37, author = {Hoang, Duc A. and Uehara, Ryuhei}, title = {{Sliding Tokens on a Cactus}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {37:1--37:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.37}, URN = {urn:nbn:de:0030-drops-68074}, doi = {10.4230/LIPIcs.ISAAC.2016.37}, annote = {Keywords: reconfiguration problem, token sliding, independent set, cactus} }

Document

**Published in:** LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

We investigate a silhouette puzzle that is recently developed based on the golden ratio. Traditional silhouette puzzles are based on a simple tile. For example, the tangram is based on isosceles right triangles; that is, each of seven pieces is formed by gluing some identical isosceles right triangles. Using the property, we can analyze it by hand, that is, without computer. On the other hand, if each piece has no special property, it is quite hard even using computer since we have to handle real numbers without numerical errors during computation. The new silhouette puzzle is between them; each of seven pieces is not based on integer length and right angles, but based on golden ratio, which admits us to represent these seven pieces in some nontrivial way. Based on the property, we develop an algorithm to handle the puzzle, and our algorithm succeeded to enumerate all convex shapes that can be made by the puzzle pieces.
It is known that the tangram and another classic silhouette puzzle known as Sei-shonagon chie no ita can form 13 and 16 convex shapes, respectively. The new puzzle, Nana-kin-san puzzle, admits to form 62 different convex shapes.

Takashi Horiyama, Ryuhei Uehara, and Haruo Hosoya. Convex Configurations on Nana-kin-san Puzzle. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{horiyama_et_al:LIPIcs.FUN.2016.20, author = {Horiyama, Takashi and Uehara, Ryuhei and Hosoya, Haruo}, title = {{Convex Configurations on Nana-kin-san Puzzle}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-005-7}, ISSN = {1868-8969}, year = {2016}, volume = {49}, editor = {Demaine, Erik D. and Grandoni, Fabrizio}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.20}, URN = {urn:nbn:de:0030-drops-58730}, doi = {10.4230/LIPIcs.FUN.2016.20}, annote = {Keywords: silhouette puzzles, nana-kin-san puzzle, enumeration algorithm, convex polygon} }