Tableaux de young et solitaire bulgare

RésuméThe following game is called in [M. Gardner, Mathematical Games, Scientific American 249 (1983), 8–13] the Bulgarian Solitar. Initially, we are given n stones disposed in several heaps. A move consists of removing exactly one stone from each heap and forming a new heap. By repeating this operation a sufficiently large number of times we eventually obtain a periodic sequence of positions. It has been proved by J. Brandt that if n is a triangular number, i.e., n = k(k + 1)2, then the final period is one. D. Knuth has conjectured that for triangular n the length of the game (i.e., the number of moves before the final position is reached) is at most k(k − 1). Our main result is the proof of a generalization of this conjecture for any integer n

Random Bulgarian solitaire

We consider a stochastic variant of the game of Bulgarian solitaire [M. Gardner (1983), Sci. Amer. 249, 12-21]. For the stationary measure of the random Bulgarian solitaire, we prove that most of its mass is concentrated on (roughly) triangular configurations of certain type.Comment: 28 pages, 4 figures; revised version - to appear in Random Structures and Algorithm