Next Article in Journal
Standard and Specific Compression Techniques for DNA Microarray Images
Next Article in Special Issue
Any Monotone Function Is Realized by Interlocked Polygons
Previous Article in Journal
Compression-Based Tools for Navigation with an Image Database
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:

How to Solve the Torus Puzzle

Department of Computer Science, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma 376-8515, Japan
Kiryu High School, 1-39 Miharacho, Kiryu, Gunma 376-0025, Japan
Author to whom correspondence should be addressed.
Algorithms 2012, 5(1), 18-29;
Received: 27 December 2011 / Revised: 30 December 2011 / Accepted: 30 December 2011 / Published: 13 January 2012
(This article belongs to the Special Issue Puzzle/Game Algorithms)


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.
Keywords: torus puzzle; 15 puzzle torus puzzle; 15 puzzle

Share and Cite

MDPI and ACS Style

Amano, K.; Kojima, Y.; Kurabayashi, T.; Kurihara, K.; Nakamura, M.; Omi, A.; Tanaka, T.; Yamazaki, K. How to Solve the Torus Puzzle. Algorithms 2012, 5, 18-29.

AMA Style

Amano K, Kojima Y, Kurabayashi T, Kurihara K, Nakamura M, Omi A, Tanaka T, Yamazaki K. How to Solve the Torus Puzzle. Algorithms. 2012; 5(1):18-29.

Chicago/Turabian Style

Amano, Kazuyuki, Yuta Kojima, Toshiya Kurabayashi, Keita Kurihara, Masahiro Nakamura, Ayaka Omi, Toshiyuki Tanaka, and Koichi Yamazaki. 2012. "How to Solve the Torus Puzzle" Algorithms 5, no. 1: 18-29.

Article Metrics

Back to TopTop