🔢 mathematics

A short survey the game Bulgarian solitaire and related games

This paper provides a concise survey of Bulgarian solitaire, a dynamical system on integer partitions that converges to a unique fixed point when the total is a triangular number, along with an overview of several variations of the game.

Original authors: Romeo Meštrović

Published 2026-07-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Romeo Meštrović

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Great Card Shuffle: A Journey into Mathematical Patterns

Imagine a world where numbers aren't just cold, static symbols on a page, but living, breathing characters that dance, jump, and rearrange themselves according to strict rules. This is the playground of combinatorics, a branch of mathematics that studies how things can be counted, arranged, and combined. At the heart of this paper lies a specific type of puzzle called integer partitioning. Think of a partition as taking a pile of identical LEGO bricks and snapping them together into different-sized towers. If you have 10 bricks, you could make one tall tower of 10, or two towers of 5, or a chaotic mix of 4, 3, 2, and 1. The question mathematicians love to ask is: "If we keep changing the rules of how we build these towers, will they ever stop moving, or will they just spin in circles forever?"

This curiosity isn't just about games; it helps us understand the hidden order in chaotic systems, from how traffic flows to how atoms arrange themselves. The paper we are exploring dives into a famous game called Bulgarian Solitaire, a simple card game that turns out to be a deep mathematical mystery. It asks: If you shuffle a deck of cards according to a very specific, repetitive rule, do the piles eventually settle into a perfect, unchanging pattern, or do they get stuck in an endless loop? The answer depends entirely on the total number of cards you start with, revealing a beautiful connection between simple arithmetic and complex cycles.

The Game: A Card Trick with a Twist

Imagine you are sitting at a table with a deck of cards, but instead of playing poker, you are playing a game of pure logic called Bulgarian Solitaire. You start by dealing your cards into several piles of any size you like. Maybe you have one pile of 5 cards, another of 3, and a tiny one with just 2. Now, here is the magic move: you reach into every single pile, pull out exactly one card from each, and gather those cards to form a brand new pile.

If you had 5 piles, you now have 5 new cards in your hand, which become a 6th pile. The old piles are now one card smaller. If a pile was only 1 card tall, it disappears entirely because you took its only card away. You repeat this move over and over again. It's like a rhythmic dance where the piles shrink, grow, and swap places. The big question is: Does this dance ever stop? Do the piles eventually arrange themselves into a specific, unchanging formation, or do they just keep shuffling forever?

The Magic Number: Triangular Numbers

The paper reveals that the answer depends on a special kind of number called a triangular number. You can make these by stacking dots in a triangle: 1 dot, then 3 dots (1+2), then 6 dots (1+2+3), then 10 (1+2+3+4), and so on. These numbers are the "golden tickets" of the game.

If you start with a number of cards that is a triangular number (like 1, 3, 6, 10, 15, etc.), the game has a guaranteed happy ending. No matter how messy your starting piles are, if you keep playing, the cards will eventually settle into a perfect, stable pattern. This pattern is always the same: piles of sizes 1, 2, 3, up to $k$ , where $k$ is the number that makes your total a triangular number. For example, if you have 10 cards (which is $1+2+3+4$ ), the game will eventually stop with piles of 1, 2, 3, and 4 cards. The mathematicians call this the unique fixed point, meaning once you reach it, the next move just recreates the same piles, and the dance stops.

The Loop: When the Number Isn't Perfect

But what happens if you don't have a triangular number of cards? Say you have 11 cards instead of 10. The paper explains that the game will never settle into a single, unchanging pattern. Instead, the piles will get stuck in a cycle. Imagine a carousel that never stops; the piles will keep changing, but after a certain number of moves, they will return to a configuration they've seen before, and the whole sequence will repeat forever.

The paper details exactly how these cycles work. It turns out that for any number of cards, the game is predictable. If the number isn't triangular, the piles will bounce between a few specific shapes in a loop. The authors describe a way to list every single possible loop for any deck size, proving that the chaos is actually very organized. It's like knowing that a spinning top will wobble in a specific circle rather than flying off randomly.

The History: From Trains to Math Journals

This game has a colorful backstory. While it was popularized in 1983 by the famous math columnist Martin Gardner, who gave it the catchy name "Bulgarian Solitaire," the story goes that it was actually discovered around 1980 by a mathematician named Konstantin Oskolkov. Legend has it that Oskolkov heard about the problem on a train ride in Russia. When he told his colleagues, one number theorist was so excited (or perhaps overwhelmed) by the puzzle that he reportedly ran to his office, locked the door, and didn't come out until he had solved it!

The paper notes that by 1981, several mathematicians had already published solutions proving that the triangular number case always leads to that single stable pattern. In 1982, a mathematician named Brandt did the heavy lifting to figure out exactly what happens when the number isn't triangular, mapping out all the possible loops. Since then, other researchers have tried to prove these facts in different ways, using tools like computer programs and different mathematical theories, but they all agree on the core result: the game is a perfect example of how simple rules can create complex, predictable patterns.

Variations: Spinning the Wheel

The paper also takes a quick look at how this game has inspired many "cousins." Mathematicians love to tweak the rules to see what happens. Some versions let you pick more than one card from a pile, or they introduce randomness, like rolling a die to decide how many cards to move. These variations, named things like "Austrian Solitaire" or "Montreal Solitaire," are still being studied today. They help researchers understand how small changes in the rules can completely change the behavior of the system, sometimes making the cycles longer or the patterns more complex.

The Bottom Line

In the end, this paper is a survey—a friendly tour guide—through the world of Bulgarian Solitaire and its many relatives. It confirms that for triangular numbers, the game always finds a peaceful, unique resting place. For all other numbers, it finds a rhythmic, repeating cycle. The authors don't just say this happens; they have mathematically proved it, showing exactly what those cycles look like and how many there are. It's a reminder that even in a game of shuffling cards, there is a deep, unshakeable order waiting to be discovered.

Technical Summary of "A Short Survey of the Game Bulgarian Solitaire and Related Games"

Problem Statement
The paper addresses the mathematical properties of "Bulgarian solitaire," a dynamical system defined on the set of integer partitions of a positive integer $N$ . The system operates via a specific shift operation $T$ : given a partition $\lambda = (\lambda_1, \lambda_2, \dots, \lambda_l)$ of $N$ , the operation $T(\lambda)$ produces a new partition consisting of the parts $\lambda_1-1, \lambda_2-1, \dots, \lambda_l-1$ and the integer $l$ (the number of parts), with any resulting zeros ignored. The core problem involves characterizing the behavior of the sequence $\lambda, T(\lambda), T^2(\lambda), \dots$ . Specifically, the paper investigates under what conditions this sequence converges to a fixed point versus entering a cycle, and seeks to classify all $T$ -cyclic partitions (partitions $\mu$ where $T^i(\mu) = \mu$ for some $i \ge 1$ ) for arbitrary $N$ .

Methodology and Formal Framework
The paper adopts a formal combinatorial approach rooted in the theory of integer partitions, originally founded by Leonhard Euler. The methodology involves:

Formal Definition: Defining the game state space as the set of partitions of $N$ and the transition function $T$ .
Cycle Characterization: Utilizing the fact that the state space is finite to establish that the system must eventually enter a cycle.
Structural Analysis: Relying on the work of Brandt [9], who characterized cycles using a specific set $M_n$ of sequences defined by cardinality conditions. The paper synthesizes results from Brandt, Akin and Davis [1], and Etienne [16] to describe the structure of cyclic partitions.
Historical Synthesis: The paper functions as a survey, compiling and contextualizing historical developments, proofs, and variations of the game from 1980 through 2016, referencing key contributions from Oskolkov, Toom, Bojanov, Eriksson, Gardner, Knuth, and others.

Key Contributions and Results
The primary contribution of the paper is a consolidated survey of the theoretical landscape of Bulgarian solitaire and its variants. Key results presented include:

Brandt's Equilibrium Theorem: For a triangular number $N = 1 + 2 + \dots + k$ , the system possesses a unique fixed point (a cycle of length 1) given by the partition $(k, k-1, \dots, 1)$ . Starting from any initial configuration, the system converges to this state.
Characterization of Cyclic Partitions: For arbitrary $N$ , the paper presents the theorem (attributed to Etienne [16] and derived from Brandt [9]) stating that a partition $\lambda$ of $N = 1 + 2 + \dots + k + r$ (where $0 \le r \le k$ ) is $T$ -cyclic if and only if it takes the form:
$\lambda = (k + \delta_k, k-1 + \delta_{k-1}, \dots, 1 + \delta_1, \delta_0)$
where each $\delta_i \in \{0, 1\}$ and $\sum_{i=0}^k \delta_i = r$ .
Conjecture Resolution: The paper notes the resolution of D. Knuth's conjecture (that the number of moves to reach the fixed point for triangular $N$ is at most $k(k-1)$ ) by Igusa [30], and the generalization of this bound for arbitrary $N$ by Etienne [16].
Variants and Generalizations: The survey catalogs numerous variations, including:
- Austrian solitaire (Akin and Davis, 1985)
- Montreal solitaire (Cannings and Haigh, 1992)
- Carolina solitaire (Andreev, 1997)
- Random Bulgarian solitaire (Popov, 2003)
- Two-handed Bulgarian solitaire (Bancroff, 2004)
- $\sigma$ -Bulgarian solitaire (Olson, 2016), which allows picking multiple cards from a single pile.
Related Counting Problems: The paper highlights a connection between Bulgarian solitaire and a periodicity problem regarding integer sequences defined by $a_n = \#\{i : 1 \le i \le n-1 \text{ and } a_i + i < n\}$ , noting that the solution to this problem was provided by Bojanov [8].

Significance and Scope
The paper positions itself as a "short survey" intended to provide a comprehensive overview of the game's history, mathematical formulation, and subsequent research. It emphasizes the game's role as a bridge between recreational mathematics (popularized by Martin Gardner in 1983) and rigorous combinatorial game theory. The significance lies in its aggregation of disparate results—ranging from proofs of convergence to the classification of cycles and the exploration of stochastic and deterministic variants—into a single reference. The author modestly frames the work as a summary of existing literature, including all twenty references from Hopkins' 2012 review plus fourteen additional sources, rather than presenting new primary theorems regarding the solitaire game itself.