The Bulgarian solitaire and the mathematics around it

TL;DR

The paper explores the Bulgarian solitaire card game, its mathematical properties, stable configurations, and related open problems, highlighting its educational and theoretical significance in combinatorics.

Contribution

It provides a comprehensive overview of the mathematical analysis of Bulgarian solitaire, including its behavior on triangular numbers and related combinatorial problems.

Findings

Game reaches stable configuration for triangular numbers

Stable piles are of sizes 1, 2, ..., k

Discussion of open mathematical problems related to the game

Abstract

The Bulgarian solitaire is a mathematical card game played by one person. A pack of n cards is divided into several decks (or "piles"). Each move consists of the removing of one card from each deck and collecting the removed cards to form a new deck. The game ends when the same position occurs twice. It has turned out that when n=k(k+1)/2 is a triangular number, the game reaches the same stable configuration with size of the piles 1,2,...,k. The purpose of the paper is to tell the (quite amusing) story of the game and to discuss mathematical problems related with the Bulgarian solitaire. The paper is dedicated to the memory of Borislav Bojanov (1944-2009), a great mathematician, person, and friend, and one of the main protagonists in the story of the Bulgarian solitaire.