Comment faire un demi-tour dans la Tour d'Hanoï Cyclique

Abstract

It is shown that, in the cyclic tower of Hanoi with $4$ pegs and $N$ disks, there exists an initial configuration of disks and a valid sequence of moves, after which the configuration has made a half-turn, where the number of disk moves grows in $O(Z^N)$, where $Z\simeq 1.69562$ is the unique real root of the equation $Z^3=Z^2+2$. This result is consistent with some numerical experiments and conjectures made by Paul Zimmermann in 2017. Then, the optimality of the number of disk moves obtained by this process is discussed.