Editing Beggar-my-neighbour (section)

== Game theory ==
A longstanding question in [[combinatorial game theory]] asks whether there is a game of beggar-my-neighbour that goes on forever.  This can happen only if the game is eventually periodic—that is, if it eventually reaches some [[state (computer science)|state]] it has been in before.  Some smaller decks of cards have infinite games, such as [[Camicia (Card Game) | Camicia]],<ref>Alessandro Gentilini, [https://github.com/matthewmayer/beggarmypython/pull/5#issue-1617788484 I found that Camicia was declared non terminating]. Retrieved 2023-08-07</ref> while others do not. [[John Horton Conway|John Conway]] once listed this among his anti-[[Hilbert problems]],<ref>
{{cite encyclopedia |url= https://archive.org/details/moregamesofnocha0000unse |title= Unsolved Problems in Combinatorial Games |format= PDF |last1= Guy |first1= Richard K. |author-link1= Richard K. Guy |last2= Nowakowski |first2= Richard J. <!-- |author-link2=Richard J. Nowakowski --> |encyclopedia= More Games of No Chance |isbn= 0521808324 |series= MSRI Publications |volume= 42 |date= 25 November 2002 <!-- wrong!? |date= 24 January 2008 --> |access-date= 2018-12-03 |publisher= [[Cambridge University Press]]
 <!-- more details at http://library.msri.org/books/Book42/index.html and https://www.amazon.com/Mathematical-Sciences-Research-Institute-Publications/dp/0521155630 --> |quote= This problem reappears periodically. It was one of Conway’s ‘anti-Hilbert problems’ about 40 years ago, but must have suggested itself to players of the game over the several centuries of its existence. |url-access= registration }}</ref>
open questions whose pursuit should emphatically ''not'' drive the future of mathematical research.

A non-terminating game was first found by Brayden Casella and reported on 10 February 2024.<ref>{{Citation |last=Casella |first=Brayden |title=A Non-Terminating Game of Beggar-My-Neighbor |date=2024-03-19 |url=http://arxiv.org/abs/2403.13855 |access-date=2024-03-23 |last2=Anderson |first2=Philip M. |last3=Kleber |first3=Michael |last4=Mann |first4=Richard P. |last5=Nessler |first5=Reed |last6=Rucklidge |first6=William |last7=Williams |first7=Samuel G. |last8=Wu |first8=Nicolas}}</ref><ref name=":0" /> The cyclic game begins {{Code|---K---Q-KQAJ-----AAJ--J--}} and {{Code|----------Q----KQ-J-----KA}}. The longest terminating game known is 1164 tricks / 8344 cards, found by Reed Nessler.<ref name=":0">
{{cite web | url= https://richardpmann.com/beggar-my-neighbour-records.html | title= Known Historical Beggar-My-Neighbour Records |author=  Richard P Mann |access-date= 2024-02-10 }}</ref> 

There are <math>\approx6.54\cdot10^{20}</math> possible combinations of beggar-my-neighbour.<ref>Remy, [https://math.stackexchange.com/a/2688359/1184658 Beggar-my-neighbour possible games]. Retrieved 2023-08-07</ref>