# Magnetic Towers of Hanoi and their Optimal Solutions - Mathematics > Combinatorics

Magnetic Towers of Hanoi and their Optimal Solutions - Mathematics > Combinatorics - Download this document for free, or read online. Document in PDF available to download.

Abstract: The Magnetic Tower of Hanoi puzzle - a modified -base 3- version of theclassical Tower of Hanoi puzzle as described in earlier papers, is actually asmall set of independent sister-puzzles, depending on the -pre-coloring-combination of the tower-s posts. Starting with Red facing up on a Source post,working through an Intermediate - colored or Neutral post, and ending Bluefacing up on a Destination post, we identify the different pre-coloringcombinations in S,I,D order. The Tower-s pre-coloring combinations are{R,B,B - R,R,B ; R,B,N - N,R,B ; N,B,N - N,R,N ; R,N,B ;R,N,N - N,N,B ; N,N,N}. In this paper we investigate thesesister-puzzles, identify the algorithm that optimally solves each pre-coloredpuzzle, and prove its Optimality. As it turns out, five of the six algorithms,challenging on their own, are part of the algorithm solving the -natural-, FreeMagnetic Tower of Hanoi puzzle N,N,N. We start by showing that the N-diskColored Tower R,B,B - R,R,B is solved by 3^N - 1-2 moves. Defining-Algorithm Duration- as the ratio of number of algorithm-moves solving thepuzzle to the number of algorithm-moves solving the Colored Tower, we find theDuration-Limits for all sister-puzzles. In the order of the list above they are{1 ; 10-11 ; 10-11 ; 8-11 ; 7-11 ; 20-33}. Thus, the Duration-Limitof the Optimal Algorithm solving the Free Magnetic Tower of Hanoi puzzle is20-33 or 606 0-00. On the road to optimally solve this colorful Magneticpuzzle, we hit other -forward-moving- puzzle-solving algorithms. Overall welooked at 10 pairs of integer sequences. Of the twenty integer sequences, fiveare listed in the On-line Encyclopedia of Integer Sequences, the other fifteen- not yet. The large set of different solutions is a clear indication to thefreedom-of-wondering that makes this Magnetic Tower of Hanoi puzzle socolorful.

Author: Uri Levy

Source: https://arxiv.org/