Editing Chess (section)

===Mathematics===
{{See also|Mathematical chess problem|Solving chess}}
The game structure and nature of chess are related to several branches of mathematics. Many [[combinatorics|combinatorical]] and [[topology|topological]] problems connected to chess, such as the [[knight's tour]] and the [[eight queens puzzle]], have been known for hundreds of years.

[[File:Knight's tour.svg|thumb|upright 0.8|Mathematicians [[Euler]], [[Adrien-Marie Legendre|Legendre]], [[de Moivre]], and [[Vandermonde]] studied the [[knight's tour]].]]
The number of legal positions in chess is estimated to be {{thinspace|(4.59|±|0.38)|×|10<sup>44</sup>}} with a 95% confidence level,<ref>{{cite web |title=Chess Position Ranking |author=John Tromp |website=[[GitHub]] | year=2021 | url=https://github.com/tromp/ChessPositionRanking | url-status=live|archive-url=https://web.archive.org/web/20210808152713/https://github.com/tromp/ChessPositionRanking |archive-date=8 August 2021 }}</ref> with a [[game-tree complexity]] of approximately 10<sup>123</sup>. The game-tree complexity of chess was first calculated by [[Claude Shannon]] as 10<sup>120</sup>, a number known as the [[Shannon number]].<ref>{{cite web|title=Evolution of the Chess Robot: Brute force Wins|first=Jonathan|last=Babb|publisher=Massachusetts Institute of Technology|date=January 10, 1996|url=http://groups.csail.mit.edu/cag/pub/papers/jbabb-area.ps.Z|access-date=21 February 2023|archive-date=21 February 2023|archive-url=https://web.archive.org/web/20230221064724/http://groups.csail.mit.edu/cag/pub/papers/jbabb-area.ps.Z|url-status=live}}</ref> An average position typically has thirty to forty possible moves, but there may be as few as zero (in the case of checkmate or stalemate) or (in a constructed position) as many as 218.<ref>{{cite web |url=http://www.chessbox.de/Compu/schachzahl2_e.html |archive-url=https://web.archive.org/web/20070613072827/http://www.chessbox.de/Compu/schachzahl2_e.html |archive-date=13 June 2007 |publisher=ChessBox.de |title=The biggest Number of simultaneous possible legal Moves |url-status=dead}}</ref>

In 1913, [[Ernst Zermelo]] used chess as a basis for his theory of game strategies, which is considered one of the predecessors of [[game theory]].<ref>Zermelo, Ernst (1913), Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, Proceedings of the Fifth International Congress of Mathematicians 2, 501–04. Cited from Eichhorn, Christoph: Der Beginn der Formalen Spieltheorie: Zermelo (1913), [http://www.mathematik.uni-muenchen.de/~spielth/artikel/Zermelo.pdf Uni-Muenchen.de] {{webarchive|url=https://web.archive.org/web/20070612134609/http://www.mathematik.uni-muenchen.de/~spielth/artikel/Zermelo.pdf |date=12 June 2007 }}. Retrieved 23 March 2007.</ref> [[Zermelo's theorem (game theory)|Zermelo's theorem]] states that it is possible to [[Solving chess|solve chess]], i.e. to determine with certainty the outcome of a perfectly played game (either White can force a win, or Black can force a win, or both sides can force at least a draw).<ref>{{cite web |url=http://www.gap-system.org/~history/Projects/MacQuarrie/Chapters/Ch4.html |title=Fundamentals |work=gap-system.org |url-status=dead |archive-url=https://web.archive.org/web/20110607183513/http://www.gap-system.org/~history/Projects/MacQuarrie/Chapters/Ch4.html |archive-date=7 June 2011}}</ref> With 10<sup>43</sup> legal positions in chess, however, it will take an impossibly long time to compute a perfect strategy with any feasible technology.<ref>{{cite book |title=Encyclopedia of Computer Science and Technology: Volume 8 |chapter=Games |page=394 |isbn=978-0-8247-2258-6 |publisher=CRC Press |year=1977}}</ref>

====Applied mathematics====
A novel methodology in [[steganography]] explores the use of chess-based covers (such as puzzles, chess problems, game reports, training documents, news articles, etc.) for concealing data within a selection of {{Chessgloss|move|moves}}, each hiding some [[bit]]s.<ref>{{cite journal |last1=Desoky |first1=Abdelrahman |last2=Younis |first2=Mohamed |title=Chestega: chess steganography methodology |journal=Security and Communication Networks |date=November 2009 |volume=2 |issue=6 |pages=555–566 |doi=10.1002/sec.99 |doi-access=free }}</ref><ref>{{cite journal |last1=Hernandez-Castro |first1=Julio C. |last2=Blasco-Lopez |first2=Ignacio |last3=Estevez-Tapiador |first3=Juan M. |last4=Ribagorda-Garnacho |first4=Arturo |title=Steganography in games: A general methodology and its application to the game of Go |journal=Computers & Security |date=February 2006 |volume=25 |issue=1 |pages=64–71 |doi=10.1016/j.cose.2005.12.001 }}</ref> Several proof-of-concept projects have been developed that convert text or files into [[binary code]], which is then converted into a series of legal chess moves, that can then be [[Decryption|decrypted]] and downloaded.<ref>{{Cite web |last=Mukherjee |first=Nayanika |date=15 September 2024 |title='I turned Lichess into Google Drive': 18yo shows how to store files, secret messages in a game of chess |url=https://indianexpress.com/article/puzzles-and-games/info/chess-games-cloud-storage-creative-coding-cryptography-brainteasers-9567020/ |access-date=28 January 2025 |newspaper=[[The Indian Express]]}}</ref>

[[Correspondence chess]] has been historically suspected of being a potential steganographic medium. [[Melville Davisson Post]] documented a [[chess problem]] that was used to create a pictorial [[cipher]] during [[World War I]].<ref>{{Cite magazine |last=Post |first=Melville Davisson |author-link=Melville Davisson Post |date=January 1918 |title=German War Ciphers |hdl=2027/uiug.30112077173703?urlappend=%3Bseq=626%3Bownerid=13510798903194481-672 |hdl-access=free |magazine=[[Everybody's Magazine]] |volume=38 |number=1 |pages=28–34 }}</ref><ref>{{Cite magazine |last=Oberhaus |first=Daniel |date=25 March 2017 |title=The Spy Who Checkmated Me: Why Postal Chess Was Banned During Wartime |url=https://www.vice.com/en/article/the-spy-who-checkmated-me-why-postal-chess-was-banned-during-wartime/ |access-date=28 January 2025 |magazine=[[Vice (magazine)|Vice]]}}</ref> During [[World War II]], extensive [[postal censorship]] was imposed on [[military personnel]] from the United States and Canada that made playing correspondence chess impossible, arising from suspicion that chess could be used to send secret messages to the enemies.<ref>{{Cite magazine |last=Dubois |first=Joan |date=September 1991 |title=Golden Knights in Review |url=https://uscf1-nyc1.aodhosting.com/CL-AND-CR-ALL/CL-ALL/1991/1991_09.pdf#page=40 |magazine=[[Chess Life]] |pages=40–45 |via=[[US Chess Federation]] |volume=46 |number=9}}</ref>