On the Footsteps to Generalized Tower of Hanoi Strategy

TL;DR

This paper proves the optimality of a recursive algorithm for the four-peg Tower of Hanoi puzzle using Bifurcation Theorem and induction, and discusses extending these results to generalized strategies.

Contribution

It provides a formal proof of the Frame-Stewart algorithm's optimality for Reve's puzzle and explores the potential for generalized solutions using Bifurcation Theorem.

Findings

Proved the optimality of Frame-Stewart algorithm for Reve's puzzle.

Analyzed strategies for solving multi-peg Tower of Hanoi puzzles.

Discussed the potential for generalized solutions using Bifurcation Theorem.

Abstract

In this paper, our aim is to prove that our recursive algorithm to solve the "Reve's puzzle" (four- peg Tower of Hanoi) is the optimal solution according to minimum number of moves. Here we used Frame's five step algorithm to solve the "Reve's puzzle", and proved its optimality analyzing all possible strategies to solve the problem. Minimum number of moves is important because no one ever proved that the "presumed optimal" solution, the Frame-Stewart algorithm, always gives the minimum number of moves. The basis of our proof is Bifurcation Theorem. In fact, we can solve generalized "Tower of Hanoi" puzzle for any pegs (three or more pegs) using Bifurcation Theorem. But our scope is limited to the "Reve's puzzle" in this literature, but lately, we would discuss how we can reach our final destination, the Generalized Tower of Hanoi Strategy. Another important point is that we have used only induction method to prove all the results throughout this literature. Moreover, some simple theorems and lemmas are derived through logical perspective or consequence of induction method. Lastly, we will try to answer about uniqueness of solution of this famous puzzle.