Bulgarian solitaire

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, $$N=1+2+\cdots+k$$ for some $$k$$), then it is known that Bulgarian solitaire will reach a stable configuration in which the sizes of the piles are $$1,2,\ldots, k$$. 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.