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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.
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)
PDF [191 KB, uploaded 13 January 2012]


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
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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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.

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