On the treewidth of Hanoi graphs

The objective of the well-known Towers Hanoi puzzle is to move a set disks one at time from pegs another, while keeping sorted on each peg. We propose an adversarial variation in which first player forbids states puzzle, and second must then convert randomly-selected state another without passing through forbidden states. Analyzing this version raises question treewidth graphs. find number exactly for three-peg puzzles provide nearly-tight asymptotic bounds larger numbers pegs.