Solving the Tower of Hanoi with Random Moves

TL;DR

This paper derives exact formulas for the expected number of random moves needed to solve various Tower of Hanoi variants with 3 pegs and n disks, using probabilistic and electrical network analysis techniques.

Contribution

It provides new exact formulae for expected move counts in Tower of Hanoi variants and introduces an alternative proof linking random walk commute times with electrical network transformations.

Findings

Exact formulas for expected moves in Tower of Hanoi variants

Connection between random walk commute times and electrical network analysis

Alternative proof using delta-to-wye transformation

Abstract

We prove the exact formulae for the expected number of moves to solve several variants of the Tower of Hanoi puzzle with 3 pegs and n disks, when each move is chosen uniformly randomly from the set of all valid moves. We further present an alternative proof for one of the formulae that couples a theorem about expected commute times of random walks on graphs with the delta-to-wye transformation used in the analysis of three-phase AC systems for electrical power distribution.