A short survey the game Bulgarian solitaire and related games

Let $N$ be an arbitrary positive integer and let $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_l)$ be a partition of $N$ of length $l$, i.e., $\sum_{i=1}^l\lambda_i= N$ with parts $\lambda_1\ge \lambda_2\ldots \lambda_l\ge 1$. Define $T(\lambda)$ as the partition of $N$ with parts $l,\lambda_1-1\lambda_2-1,\ldots \lambda_l-1$,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 $$\lambda, T(\lambda), T^2(\lambda),\ldots .$$ 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.