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Algorithms
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Puzzle/Game Algorithms
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Puzzle/Game Algorithms

A special issue of Algorithms (ISSN 1999-4893).

Deadline for manuscript submissions: closed (15 March 2012) | Viewed by 60019

Special Issue Editor

Prof. Dr. Ryuhei Uehara

E-Mail Website
Guest Editor
School of Information Science (JAIST), 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan
Interests: computational complexity; graph theory; algorithms
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the society of theoretical computer science, games and puzzles have been widely investigated since they sometimes require deep insight from the viewpoint of mathematics. From the viewpoint of computation, John Conway’s Game of Life plays an important role as a computation model, and the pebble games are used to formulate the computational complexity classes. On the other hand, from the viewpoint of algorithms, combinatorial game theory has also been in fashion since Bouton’s analysis of the game of Nim, more than one hundred years ago. This special issue will focus on research papers that describe puzzle/game algorithms. The topics include, but not limited to:

  • algorithms for generating puzzle instances
  • algorithms for solving games/puzzles
  • computational complexity of games/puzzles
  • mathematical analysis of games/puzzles
  • recreational mathmatics
  • winning strategies of games/puzzles

Prof. Dr. Ryuhei Uehara
Guest Editor

Keywords

  • combinatorial game theory
  • recreational mathematics
  • computational complexity of games/puzzles
  • algorithms for games/puzzles
  • mathematics in games/puzzles

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Published Papers (6 papers)

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4034 KiB  
Article
Imaginary Cubes and Their Puzzles
by Hideki Tsuiki
Algorithms 2012, 5(2), 273-288; https://doi.org/10.3390/a5020273 - 9 May 2012
Cited by 2 | Viewed by 8575
Abstract
Imaginary cubes are three dimensional objects which have square silhouette projections in three orthogonal ways just as a cube has. In this paper, we study imaginary cubes and present assembly puzzles based on them. We show that there are 16 equivalence classes of [...] Read more.
Imaginary cubes are three dimensional objects which have square silhouette projections in three orthogonal ways just as a cube has. In this paper, we study imaginary cubes and present assembly puzzles based on them. We show that there are 16 equivalence classes of minimal convex imaginary cubes, among whose representatives are a hexagonal bipyramid imaginary cube and a triangular antiprism imaginary cube. Our main puzzle is to put three of the former and six of the latter pieces into a cube-box with an edge length of twice the size of the original cube. Solutions of this puzzle are based on remarkable properties of these two imaginary cubes, in particular, the possibility of tiling 3D Euclidean space. Full article
(This article belongs to the Special Issue Puzzle/Game Algorithms)
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117 KiB  
Article
A Polynomial-Time Reduction from the 3SAT Problem to the Generalized String Puzzle Problem
by Chuzo Iwamoto, Kento Sasaki and Kenichi Morita
Algorithms 2012, 5(2), 261-272; https://doi.org/10.3390/a5020261 - 13 Apr 2012
Cited by 3 | Viewed by 7497
Abstract
A disentanglement puzzle consists of mechanically interlinked pieces, and the puzzle is solved by disentangling one piece from another set of pieces. A string puzzle consists of strings entangled with one or more wooden pieces. We consider the generalized string puzzle problem whose [...] Read more.
A disentanglement puzzle consists of mechanically interlinked pieces, and the puzzle is solved by disentangling one piece from another set of pieces. A string puzzle consists of strings entangled with one or more wooden pieces. We consider the generalized string puzzle problem whose input is the layout of strings and a wooden board with holes embedded in the 3-dimensional Euclidean space. We present a polynomial-time transformation from an arbitrary instance ƒ of the 3SAT problem to a string puzzle s such that ƒ is satisfiable if and only if s is solvable. Therefore, the generalized string puzzle problem is NP-hard. Full article
(This article belongs to the Special Issue Puzzle/Game Algorithms)
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298 KiB  
Article
Finding All Solutions and Instances of Numberlink and Slitherlink by ZDDs
by Ryo Yoshinaka, Toshiki Saitoh, Jun Kawahara, Koji Tsuruma, Hiroaki Iwashita and Shin-ichi Minato
Algorithms 2012, 5(2), 176-213; https://doi.org/10.3390/a5020176 - 5 Apr 2012
Cited by 23 | Viewed by 13170
Abstract
Link puzzles involve finding paths or a cycle in a grid that satisfy given local and global properties. This paper proposes algorithms that enumerate solutions and instances of two link puzzles, Slitherlink and Numberlink, by zero-suppressed binary decision diagrams (ZDDs). A ZDD is [...] Read more.
Link puzzles involve finding paths or a cycle in a grid that satisfy given local and global properties. This paper proposes algorithms that enumerate solutions and instances of two link puzzles, Slitherlink and Numberlink, by zero-suppressed binary decision diagrams (ZDDs). A ZDD is a compact data structure for a family of sets provided with a rich family of set operations, by which, for example, one can easily extract a subfamily satisfying a desired property. Thanks to the nature of ZDDs, our algorithms offer a tool to assist users to design instances of those link puzzles. Full article
(This article belongs to the Special Issue Puzzle/Game Algorithms)
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6813 KiB  
Article
An Integer Programming Approach to Solving Tantrix on Fixed Boards
by Fumika Kino and Yushi Uno
Algorithms 2012, 5(1), 158-175; https://doi.org/10.3390/a5010158 - 22 Mar 2012
Cited by 2 | Viewed by 5843
Abstract
Tantrix (Tantrix R ⃝ is a registered trademark of Colour of Strategy Ltd. in New Zealand, and of TANTRIX JAPAN in Japan, respectively, under the license of M. McManaway, the inventor.) is a puzzle to make a loop by connecting lines drawn on [...] Read more.
Tantrix (Tantrix R ⃝ is a registered trademark of Colour of Strategy Ltd. in New Zealand, and of TANTRIX JAPAN in Japan, respectively, under the license of M. McManaway, the inventor.) is a puzzle to make a loop by connecting lines drawn on hexagonal tiles, and the objective of this research is to solve it by a computer. For this purpose, we first give a problem setting of solving Tantrix as making a loop on a given fixed board. We then formulate it as an integer program by describing the rules of Tantrix as its constraints, and solve it by a mathematical programming solver to have a solution. As a result, we establish a formulation that can solve Tantrix of moderate size, and even when the solutions are invalid only by elementary constraints, we achieved it by introducing additional constraints and re-solve it. By this approach we succeeded to solve Tantrix of size up to 60. Full article
(This article belongs to the Special Issue Puzzle/Game Algorithms)
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2190 KiB  
Article
Any Monotone Function Is Realized by Interlocked Polygons
by Erik D. Demaine, Martin L. Demaine and Ryuhei Uehara
Algorithms 2012, 5(1), 148-157; https://doi.org/10.3390/a5010148 - 19 Mar 2012
Cited by 2 | Viewed by 6248
Abstract
Suppose there is a collection of n simple polygons in the plane, none of which overlap each other. The polygons are interlocked if no subset can be separated arbitrarily far from the rest. It is natural to ask the characterization of the subsets [...] Read more.
Suppose there is a collection of n simple polygons in the plane, none of which overlap each other. The polygons are interlocked if no subset can be separated arbitrarily far from the rest. It is natural to ask the characterization of the subsets that makes the set of interlocked polygons free (not interlocked). This abstracts the essence of a kind of sliding block puzzle. We show that any monotone Boolean function ƒ on n variables can be described by m = O(n) interlocked polygons. We also show that the decision problem that asks if given polygons are interlocked is PSPACE-complete. Full article
(This article belongs to the Special Issue Puzzle/Game Algorithms)
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191 KiB  
Article
How to Solve the Torus Puzzle
by Kazuyuki Amano, Yuta Kojima, Toshiya Kurabayashi, Keita Kurihara, Masahiro Nakamura, Ayaka Omi, Toshiyuki Tanaka and Koichi Yamazaki
Algorithms 2012, 5(1), 18-29; https://doi.org/10.3390/a5010018 - 13 Jan 2012
Cited by 6 | Viewed by 15806
Abstract
In this paper, we consider the following sliding puzzle called torus puzzle. In an m by n board, there are mn pieces numbered from 1 to mn. Initially, the pieces are placed in ascending order. Then they are scrambled by rotating the [...] Read more.
In this paper, we consider the following sliding puzzle called torus puzzle. In an m by n board, there are mn pieces numbered from 1 to mn. Initially, the pieces are placed in ascending order. Then they are scrambled by rotating the rows and columns without the player’s knowledge. The objective of the torus puzzle is to rearrange the pieces in ascending order by rotating the rows and columns. We provide a solution to this puzzle. In addition, we provide lower and upper bounds on the number of steps for solving the puzzle. Moreover, we consider a variant of the torus puzzle in which each piece is colored either black or white, and we present a hardness result for solving it. Full article
(This article belongs to the Special Issue Puzzle/Game Algorithms)
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