A short survey the game Bulgarian solitaire and related games

Let $N$ be an arbitrary positive integer and let $\lambda =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_l)$ be a partition of $N$ of length $l$, i.e., ${\sum }_{i=1}^l{\lambda }_i=N$ with parts ${\lambda }_1\ge {\lambda }_2\dots {\lambda }_l\ge 1$. Define $T(\lambda )$ as the partition of $N$ with parts $l,{\lambda }_1-1{\lambda }_2-1,\dots {\lambda }_l-1$,ignoring any zeros that might occur. Starting with a partition $\lambda$ of $N$, we describe Bulgarian solitaire by repeatedly applying the shift operation $T$ to obtain the sequence of partitions $$\lambda ,T(\lambda ),T^2(\lambda ),\dots \mathrm{.}$$ We say a partition $\mu$ of $N$ is $T$-cyclic if $T(\mu )=\mu$ for some $i\ge 1$. In 1982 Brandt [9] characterized all $T$-cyclic partitions for Bulgarian solitaire. Bulgarian solitaire is a dynamical system on integer partition of a positive integer $N$ which converges to a unique fixed point if $N=1+2+\cdots +k$ is a triangular number. In this paper we present a short survey of the game Bulgarian solitaire and several variations of this game.