Magnetic Towers of Hanoi and their Optimal Solutions

TL;DR

This paper explores the Magnetic Tower of Hanoi puzzle's various pre-coloring configurations, identifies optimal algorithms for each, and proves their optimality, revealing diverse solutions and complexity measures.

Contribution

It introduces and analyzes multiple sister-puzzles of the Magnetic Tower of Hanoi, providing optimal algorithms and proving their optimality for each configuration.

Findings

Optimal algorithms for each pre-colored puzzle identified

The solution for the N-disk Colored Tower uses (3^N - 1)/2 moves

The overall puzzle's optimal solution has a move ratio limit of 20/33

Abstract

The Magnetic Tower of Hanoi puzzle - a modified "base 3" version of the classical Tower of Hanoi puzzle as described in earlier papers, is actually a small 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 Blue facing up on a Destination post, we identify the different pre-coloring combinations 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 these sister-puzzles, identify the algorithm that optimally solves each pre-colored puzzle, 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", Free Magnetic Tower of Hanoi puzzle [N,N,N]. We start by showing that the N-disk Colored 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 the puzzle to the number of algorithm-moves solving the Colored Tower, we find the Duration-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-Limit of the Optimal Algorithm solving the Free Magnetic Tower of Hanoi puzzle is 20/33 or 606 0/00. On the road to optimally solve this colorful Magnetic puzzle, we hit other "forward-moving" puzzle-solving algorithms. Overall we looked at 10 pairs of integer sequences. Of the twenty integer sequences, five are 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 the freedom-of-wondering that makes this Magnetic Tower of Hanoi puzzle so colorful.