Abstract

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 $\lambda_1\ge \lambda_2\ge\ldots\ge \lambda_l\ge 1$. 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 shiftoperation $T$ to obtain the sequence of partitions $$\lambda,T(\lambda),\;T^2(\lambda),\;\ldots.$$ 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 give a short survey of this popular mathematical card game and several variations of this game.

Keywords: Bulgarian solitaire; cycle of partitions; shift operation; triangular number; partition of an integer.

DOI: 10.57016/TM-WWIF2353

Pages:  57$-$62     

Volume  XXIX ,  Issue  1 ,  2026