Sliding puzzle

<h2 id="group-theory">Group theory</h2>
<p>As a famous example of the sliding puzzle, it can be proved that the <a href="/facts/15_puzzle/kyFRwRWs">15 puzzle</a> can be represented by the <a href="/facts/Alternating_group/yethKwyo">alternating group</a> 
  
    
      
        
          A
          
            15
          
        
      
    
    {\displaystyle A_{15}}
  
,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> because the combinations of the 15 puzzle can be generated by <a href="/facts/Permutation/JSw9ukmv">3-cycles</a>. In fact, any 
  
    
      
        n
        ×
        m
      
    
    {\displaystyle n\times m}
  
 sliding puzzle with square tiles of equal size can be represented by 
  
    
      
        
          A
          
            n
            m
            −
            1
          
        
      
    
    {\displaystyle A_{nm-1}}
  
.
</p>
<h2 id="gallery">Gallery</h2>

<h2 id="examples-of-sliding-puzzles">Examples of sliding puzzles</h2>
<ul><li><a href="/facts/Fifteen_puzzle/kyFRwRWs">Fifteen puzzle</a></li>
<li><a href="/facts/Klotski/6G9oR1UL">Klotski</a></li>
<li><a href="/facts/Minus_Cube/nje8Lk3D">Minus Cube</a></li>
<li><a href="/facts/Rush_Hour_(board_game)/d6y9TSXv">Rush Hour</a></li>
<li><a href="/facts/Sokoban/T1G50XgG">Sokoban</a></li>
<li><a href="/facts/Rubik%27s_Slide/F3cBmIFu">Rubik's Slide</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Ro_(video_game)/NJ1w3PWx">Ro (video game)</a> – A rotational variation</li>
<li><a href="/facts/Rubik%27s_Cube/Nlq35AbS">Rubik's Cube</a></li></ul>

<ul><li><i>Sliding Piece Puzzles</i> (by <a href="/facts/Edward_Hordern/KFak3L4W">Edward Hordern</a>, 1986, <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-853204-0) is said to be the definitive volume on this type of puzzle.</li>
<li><i>Winning Ways</i> (by <a href="/facts/Elwyn_Ralph_Berlekamp/zUTTIlF6">Elwyn Ralph Berlekamp</a> et al., 1982, <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>)</li>
<li><i>The 15 Puzzle</i> (by <a href="/facts/Jerry_Slocum/A9GBCE0q">Jerry Slocum</a> & Dic Sonneveld, 2006, Slocum Puzzle Foundation)</li>
<li><a href="http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=4872682">US Patent 4872682</a> <a href="https://web.archive.org/web/20150603211302/http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=4872682">Archived</a> 2015-06-03 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a> - sliding puzzle wrapped on Rubik's Cube</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Mike Keith. "Vintage plastic sliding-letter puzzles" (PDF). Archived from the original (PDF) on 2014-12-17. <a href="https://web.archive.org/web/20141217140643/http://www.cs.brandeis.edu/~storer/JimPuzzles/SLIDE/CornellCrossword/KeithArticle2011.pdf" target="_blank">https://web.archive.org/web/20141217140643/http://www.cs.brandeis.edu/~storer/JimPuzzles/SLIDE/CornellCrossword/KeithArticle2011.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Beeler, Robert. "The Fifteen Puzzle: A Motivating Example for the Alternating Group" (PDF). faculty.etsu.edu/. East Tennessee State University. Archived from the original (PDF) on 2021-01-07. Retrieved 2020-12-26. <a href="https://web.archive.org/web/20210107214840/https://faculty.etsu.edu/beelerr/fifteen-supp.pdf" target="_blank">https://web.archive.org/web/20210107214840/https://faculty.etsu.edu/beelerr/fifteen-supp.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Sliding puzzle open-in-new

Sliding puzzle