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