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Bulgarian Solitaire




In the game, a pack of N cards are divided into several piles. Then for each pile, either leave it intact or, with a fixed Probability p, remove one card; collect the removed cards together to form a new pile (piles of zero size are ignored).

If p=1, the game is known as ''Bulgarian solitaire'' or ''deterministic Bulgarian solitaire'' and was introduced by Martin Gardner ; the general case with 0 is known as ''random Bulgarian solitaire'' or ''stochastic Bulgarian solitaire''. This is a Finite irreducible Markov Chain .

If N is a Triangular Number (that is, N=1+2+\ldots+k for some k), then it is known that deterministic Bulgarian solitaire will reach a stable configuration in which the size of the piles is 1,2,\ldots k. This state is reached k^2-k moves or fewer. If N is not triangular, no stable configuration exists and a Limit Cycle is reached.

In 2004 , Brazilian Statistician Serguei Popov showed that stochastic Bulgarian solitaire spends "most" of its time in a "roughly" triangular distribution.


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