Some perfect cards shuffles (French title: Quelques m\'elanges parfaits de cartes)

TL;DR

This paper investigates certain perfect card shuffles used in magic, analyzing whether repeated shuffles can return the deck to its original order, using elementary methods suitable for a broad audience.

Contribution

It provides an accessible elementary analysis of the periodicity of specific perfect card shuffles, a problem in discrete dynamical systems.

Findings

Identifies conditions under which shuffles return the deck to initial state

Calculates explicit periods for specific perfect shuffles

Demonstrates elementary approach to complex dynamical systems

Abstract

In this paper, we study some cards shuffles which are used by magicians. We focus ourselves on the possibility to hit eventually the initial state after several shuffles. This is a classical problem arising in discrete dynamical systems. The computations are performed through an elementary approach, so the paper is easily accessible. ----- Dans cet article, on \'etudie quelques m\'elanges de cartes bien connus du monde de la magie. On examine en d\'etail l'\'eventualit\'e de reconstituer le jeu de cartes initial apr\`es plusieurs m\'elanges cons\'ecutifs. Il s'agit math\'ematiquement d'un probl\`eme de syst\`emes dynamiques discrets pour lequel on recherche explicitement une p\'eriode. Les calculs \'etant pr\'esent\'es de mani\`ere \'el\'ementaire, l'article se veut accessible \`a un large public.