{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:32:54Z","timestamp":1760243574004,"version":"build-2065373602"},"reference-count":6,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2012,1,13]],"date-time":"2012-01-13T00:00:00Z","timestamp":1326412800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>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\u2019s 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.<\/jats:p>","DOI":"10.3390\/a5010018","type":"journal-article","created":{"date-parts":[[2012,1,13]],"date-time":"2012-01-13T11:31:51Z","timestamp":1326454311000},"page":"18-29","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["How to Solve the Torus Puzzle"],"prefix":"10.3390","volume":"5","author":[{"given":"Kazuyuki","family":"Amano","sequence":"first","affiliation":[{"name":"Department of Computer Science, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma 376-8515, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuta","family":"Kojima","sequence":"additional","affiliation":[{"name":"Kiryu High School, 1-39 Miharacho, Kiryu, Gunma 376-0025, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Toshiya","family":"Kurabayashi","sequence":"additional","affiliation":[{"name":"Kiryu High School, 1-39 Miharacho, Kiryu, Gunma 376-0025, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Keita","family":"Kurihara","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma 376-8515, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Masahiro","family":"Nakamura","sequence":"additional","affiliation":[{"name":"Kiryu High School, 1-39 Miharacho, Kiryu, Gunma 376-0025, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ayaka","family":"Omi","sequence":"additional","affiliation":[{"name":"Kiryu High School, 1-39 Miharacho, Kiryu, Gunma 376-0025, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Toshiyuki","family":"Tanaka","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma 376-8515, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Koichi","family":"Yamazaki","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma 376-8515, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,1,13]]},"reference":[{"key":"ref_1","unstructured":"Diaconis, P. (1988). IMS Lecture Notes-Monograph Series, Institute of Mathematical Statistics."},{"key":"ref_2","unstructured":"Bogomolny, A., and Greenwell, D. (2012, January 09). \u201cSliders\u201d\u2014Columns Cut the Knot!. Available online: http:\/\/www.maa.org\/editorial\/knot\/slider.html."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/S0747-7171(08)80001-6","article-title":"The (n2 \u2212 1)-puzzle and related relocation problems","volume":"10","author":"Ratner","year":"1990","journal-title":"J. Symb. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"397","DOI":"10.2307\/2369492","article-title":"Notes on the \u201c15\u201d Puzzle","volume":"2","author":"Johnson","year":"1879","journal-title":"Am. J. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kornhauser, D., Miller, G., and Spirakis, P. (1984, January 24\u201326). Coordinating Pebble Motion On Graphs, The Diameter Of Permutation Groups, and Applications. Proceedings of the 25th Annual Symposium on Foundations of Computer Science (SFCS \u201984), Singer Island, FL, USA.","DOI":"10.1109\/SFCS.1984.715921"},{"key":"ref_6","unstructured":"Garey, M., and Johnson, D. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Company."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/5\/1\/18\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:48:29Z","timestamp":1760219309000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/5\/1\/18"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,13]]},"references-count":6,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2012,3]]}},"alternative-id":["a5010018"],"URL":"https:\/\/doi.org\/10.3390\/a5010018","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2012,1,13]]}}}