Twin Towers of Hanoi

TL;DR

This paper investigates the Twin Towers of Hanoi problem, analyzing the minimal number of moves needed to transform initial to final configurations in coupled peg sets using group theory and automorphisms.

Contribution

It introduces bounds on optimal solution lengths for the Twin Towers of Hanoi using a group-theoretic framework involving automorphisms of rooted trees.

Findings

Established upper and lower bounds for solution lengths

Modeled configurations using Hanoi Towers group actions

Analyzed the problem through automorphisms on pairs of vertices

Abstract

In the Twin Towers of Hanoi version of the well known Towers of Hanoi Problem there are two coupled sets of pegs. In each move, one chooses a pair of pegs in one of the sets and performs the only possible legal transfer of a disk between the chosen pegs (the smallest disk from one of the pegs is moved to the other peg), but also, simultaneously, between the corresponding pair of pegs in the coupled set (thus the same sequence of moves is always used in both sets). We provide upper and lower bounds on the length of the optimal solutions to problems of the following type. Given an initial and a final position of N disks in each of the coupled sets, what is the smallest number of moves needed to simultaneously obtain the final position from the initial one in each set? Our analysis is based on the use of a group, called Hanoi Towers group, of rooted ternary tree automorphisms, which models the original problem in such a way that the configurations on N disks are the vertices at level N of the tree and the action of the generators of the group represents the three possible moves between the three pegs. The twin version of the problem is analyzed by considering the action of Hanoi Towers group on pairs of vertices.