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| In [[mathematics]] and [[game theory]], '''Bulgarian solitaire''' is a [[playing card|card]] game that was introduced by [[Martin Gardner]].
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| In the game, a pack of <math>N</math> cards is divided into several piles. Then for each pile, remove one card; collect the removed cards together to form a new pile (piles of zero size are ignored).
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| If <math>N</math> is a [[triangular number]] (that is, <math>N=1+2+\cdots+k</math> for some <math>k</math>), then it is known that Bulgarian solitaire will reach a stable configuration in which the sizes of the piles are <math>1,2,\ldots, k</math>. This state is reached in <math>k^2-k</math> moves or fewer. If <math>N</math> is not triangular, no stable configuration exists and a [[limit cycle]] is reached.
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| ==Random Bulgarian solitaire==
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| In ''random Bulgarian solitaire'' or ''stochastic Bulgarian solitaire'' a pack of <math>N</math> cards is divided into several piles. Then for each pile, either leave it intact or, with a fixed [[probability]] <math>p</math>, remove one card; collect the removed cards together to form a new pile (piles of zero size are ignored). This is a [[finite set|finite]] irreducible [[Markov chain]].
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| In 2004, [[Brazil]]ian [[probability|probabilist]] of [[Russia]]n origin [[Serguei Popov]] showed that stochastic Bulgarian solitaire spends "most" of its time in a "roughly" triangular distribution.
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| ==References==
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| *{{cite journal | author= Serguei Popov| title= Random Bulgarian solitaire| journal= [[Random Structures and Algorithms]]| year=2005| volume=27 | pages=310–330 | doi= 10.1002/rsa.20076 | issue=3}}
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| * {{cite journal | doi = 10.2307/2323643 | author = [[Ethan Akin]] and [[Morton Davis]] | title = Bulgarian solitaire | journal = [[American Mathematical Monthly]] | year = 1985 | volume = 92 | pages=237–250 | issue=4 | jstor = 2323643}}
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| [[Category:20th century card games]]
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| [[Category:Recreational mathematics]]
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| [[Category:Combinatorial game theory]]
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Latest revision as of 12:05, 25 September 2014
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