Exploring the Mathematical Properties of and Related to the Tower of Hanoi and Its Solution

The Tower of Hanoi is arguably one of the most fascinating mathematical puzzles of all time. Apparently based on the legend of a mystical Hindu temple where 64 golden discs were placed in increasing size from top to bottom, the modern version introduced by Edouard Lucas has fascinated many math-enthusiasts worldwide. Only one rule must be followed while moving all discs from the first to the last column in order to complete the puzzle - a larger disc must never be placed over a smaller one. The puzzle has encouraged countless explorations of the logic and the mathematics behind the puzzle, with more and more striking patterns becoming apparent over time. This paper will explore a basic solution of the puzzle, followed by a mathematical and combinatorial derivation of formulae relating to the minimum and maximum number of moves that can be used to solve any version of the puzzle (with a specified number of discs and columns). Some variations and applications of the puzzle and its related concepts will also be explored.