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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.
Received: 27 December 2011; in revised form: 30 December 2011 / Accepted: 30 December 2011 / Published: 13 January 2012
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 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
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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.
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.
Amano, Kazuyuki; Kojima, Yuta; Kurabayashi, Toshiya; Kurihara, Keita; Nakamura, Masahiro; Omi, Ayaka; Tanaka, Toshiyuki; Yamazaki, Koichi. 2012. "How to Solve the Torus Puzzle." Algorithms 5, no. 1: 18-29.