In mathematics and game theory, Bulgarian solitaire is a card game that was introduced by Martin Gardner.
In the game, a pack of N 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).
If N is a triangular number (that is, N = 1 + 2 + ⋯ + k for some k ), then it is known that Bulgarian solitaire will reach a stable configuration in which the sizes of the piles are 1 , 2 , … , k . This state is reached in k 2 − k moves or fewer. If N is not triangular, no stable configuration exists and a limit cycle is reached.
Random Bulgarian solitaire
In random Bulgarian solitaire or stochastic Bulgarian solitaire a pack of N cards is 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). This is a finite irreducible Markov chain.
In 2004, Brazilian probabilist of Russian origin Serguei Popov showed that stochastic Bulgarian solitaire spends "most" of its time in a "roughly" triangular distribution.
