A short survey the game Bulgarian solitaire and related games

2026/07/19 by Romeo Meštrović · 1 voice
#math.HO #math.CO #math.NT

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Abstract

Let N be an arbitrary positive integer and let λ=(λ₁, λ₂, …, λ_l) be a partition of N of length l, i.e., ∑_i=1^lλ_i= N with parts λ₁≥ λ₂… λ_l≥ 1. Define T(λ) as the partition of N with parts l,λ₁-1λ₂-1,… λ_l-1,ignoring any zeros that might occur. Starting with a partition λ of N, we describe Bulgarian solitaire by repeatedly applying the shift operation T to obtain the sequence of partitions λ, T(λ), T²(λ),… . We say a partition μ of N is T-cyclic if T(μ) = μ for some i≥ 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+⋯ +k is a triangular number. In this paper we present a short survey of the game Bulgarian solitaire and several variations of this game.

Discussions

• Romeo Me\v{s}trovi\'c: A short survey the game Bulgarian solitaire and related games https://arxiv.org/abs/2607.17194 https://arxiv.org/pdf/2607.17194 https://arxiv.org/html/2607.17194 [bsky, 0 points, 0 comments]

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