Twin Towers of Hanoi

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. 1. Towers of Hanoi and Twin Towers of Hanoi We first describe the well known Hanoi Towers Problem on n disks and 3 pegs. The n disks have different size. Allowed positions (which we call configurations) of the disks on the pegs are those in which no disk is on top of a smaller disk. An example of a configuration on 4 disks is provided in Figure 1). In a single move, the top disk from one of the pegs can be transferred to the top position on another peg as long as the newly obtained position of the disks is allowed (it is a configuration).