Editing 15 puzzle (section)

===Group theory===
The transformations of the 15 puzzle form a [[groupoid]] (not a group, as not all moves can be composed);<ref>Jim Belk (2008) [https://cornellmath.wordpress.com/2008/01/27/puzzles-groups-and-groupoids/ Puzzles, Groups, and Groupoids], The Everything Seminar</ref><ref>[http://www.neverendingbooks.org/the-15-puzzle-groupoid-1 The 15-puzzle groupoid (1)], Never Ending Books</ref><ref>[http://www.neverendingbooks.org/the-15-puzzle-groupoid-2 The 15-puzzle groupoid (2)], Never Ending Books</ref> this [[Group action (mathematics)#Generalizations|groupoid acts]] on configurations.

Because the combinations of the 15 puzzle can be generated by [[Permutation#Definition|3-cycles]], it can be proved that the 15 puzzle can be represented by the [[alternating group]] <math>A_{15}</math>.<ref>{{cite web | access-date=26 December 2020 | first1=Robert | last1=Beeler | url=https://faculty.etsu.edu/beelerr/fifteen-supp.pdf | title=The Fifteen Puzzle: A Motivating Example for the Alternating Group | publisher=East Tennessee State University | website=faculty.etsu.edu/ | archive-date=7 January 2021 | archive-url=https://web.archive.org/web/20210107214840/https://faculty.etsu.edu/beelerr/fifteen-supp.pdf | url-status=dead }}</ref> In fact, any <math>2 k - 1</math> sliding puzzle with square tiles of equal size can be represented by <math>A_{2 k - 1}</math>.