Solving the Tower of Hanoi with Random Moves

The Tower of Hanoi puzzle consists of n disks of distinct sizes distributed across 3 pegs. We will refer to a particular distribution of the disks across the pegs as a state and call it valid if on each peg disks form a pile with the disk sizes decreasing from bottom up. Since each disk can reside at one of 3 pegs, while the order of disks on each peg is uniquely defined by their sizes, the total number of valid states is 3n. At a single move it is allowed to transfer a disk from the top of one peg to the top of another peg if this results in a valid state. In the classic formulation of the Tower of Hanoi puzzle, all disk initially are located on the first peg and it is required to transfer them all to the third peg with the smallest number of moves, which is known to be 2n − 1. French mathematician Edouard Lucas invented the Tower of Hanoi puzzle in 1883 [6]. Apparently, he simultaneously created the following legend [7]: