Fibonacci-like sequences for variants of the tower of Hanoi, and corresponding graphs and gray codes

TL;DR

This paper introduces a Fibonacci-based variant of the Tower of Hanoi puzzle, revealing new properties, Gray-like codes, and algorithms involving Fibonacci substitution, with potential generalizations.

Contribution

It presents a novel Fibonacci-involved variant of Tower of Hanoi, analyzes its properties, and develops related Gray-like codes and algorithms, extending the classical puzzle.

Findings

Optimal algorithms involve Fibonacci substitution

Gray-like code on binary words without '11' factor

Extension to natural generalizations

Abstract

We modify the rules of the classical Tower of Hanoi puzzle in a quite natural way to get the Fibonacci sequence involved in the optimal algorithm of resolution, and show some nice properties of such a variant. In particular, we deduce from this Tower of Hanoi-Fibonacci a Gray-like code on the set of binary words without the factor 11, which has some properties intersting for itself and from which an iterative algorithm for the Tower of Hanoi-Fibonacci is obtained. Such an algorithm involves the Fibonacci substitution. Eventually, we briefly extend the study to some natural generalizations.