Ciril Petr

1-perfect codes in Sierpiński graphs

Sierpinski graphs S ( n , κ) generalise the Tower of Hanoi graphs—the graph S ( n , 3) is isomorphic to the graph H n of the Tower of Hanoi with n disks. A 1-perfect code (or an efficient dominating set) in a graph G is a vertex subset of G with the property that the closed neighbourhoods of its elements form a partition of V ( G ). It is proved that the graphs S ( n , κ) possess unique 1-perfect codes, thus extending a previously known result for H n . An efficient decoding algorithm is also presented. The present approach, in particular the proposed (de)coding, is intrinsically different from the approach to H n .

Metric properties of the Tower of Hanoi graphs and Stern's diatomic sequence

It is known that in the Tower of Hanoi graphs there are at most two different shortest paths between any fixed pair of vertices. A formula is given that counts, for a given vertex v, the number of vertices u such that there are two shortest u, v-paths. The formula is expressed in terms of Sterns diatomic sequence b(n) (n ≥ 0) and implies that only for vertices of degree two this number is zero. Plane embeddings of the Tower of Hanoi graphs are also presented that provide an explicit description of b(n) as the number of elements of the sets of vertices of the Tower of Hanoi graphs intersected by certain lines in the plane.

On the Frame-Stewart algorithm for the multi-peg Tower of Hanoi problem

It is proved that seven different approaches to the multi-peg Tower of Hanoi problem are all equivalent. Among them the classical approaches of Stewart and Frame from 1941 can be found.

Computational Solution of an Old Tower of Hanoi Problem

Abstract This is the amazing story of an innocent looking mathematical puzzle turning into a serious research topic in graph theory, integer sequences, and algorithms. The Tower of Hanoi and The Reves Puzzle of Lucas and Dudeney, respectively, induced a wealth of interesting mathematical and algorithmic challenges over more than a century. Although some part of the most intriguing question, the Frame-Stewart Conjecture , has recently been solved, several of the original tasks posed by Dudeney remained intractable. We present the history and theory of these questions and a computational approach which allowed us to solve a 104 years old problem of Dudeney, namely the proof of minimality of an algorithm producing paths between perfect states of the Tower of Hanoi with 5 pegs and 20 discs. Many questions about the metric properties of Hanoi graphs remain open, however, and have to be treated by analytical and computational methods in the future.

The Classical Tower of Hanoi

This chapter describes the classical TH with three pegs. In the first section, the original task to transfer a tower from one peg to another is studied in detail. We then extend our considerations to tasks that transfer discs from an arbitrary regular state to a selected peg. We further broaden our view in Section 2.4 to tasks transforming an arbitrary regular state into another regular state. For this purpose it will be useful to introduce Hanoi graphs in Section 2.3.

The Beginning of the World

The roots of mathematics go far back in history. To present the origins of the protagonist of this book, we even have to return to the Creation.

The Tower of London

The Tower of London (TL) was invented in 1982 by Shallice [379] and has received an astonishing attention in the psychology of problem solving and in neuropsychology. Just for an illustration, we point out that in the paper [208], which sets up the mathematical framework for the TL, no less than 79 references are listed! The success of the TL is due to the fact that on one hand it is an easy-to-observe psychological test tool, while on the other hand it can be applied in different situations and for numerous clinical goals. It is hence not surprising that several additional variations of the TL were proposed to which we will turn in Section 7.2. Here we only mention the Tower of Oxford introduced by G. Ward and A. Allport in [430] and named in [208, p. 2936], but which is mathematically the same puzzle as the TH without the divine rule or either the Bottleneck TH with maximal discrepancy.

Variations of the Puzzle

TH is an example of a one person game; such games are known as solitaire games. There are plenty of other mathematical solitaire games, the Icosian game, the Fifteen puzzle, and Rubik’s Cube are just a few prominent examples. Numerous variations of the TH can also be defined, some natural and some not that natural. In fact, Lucas himself in [286, p. 303] pointed out the following: “Le nombre des problemes que l’on peut se poser sur la nouvelle Tour d’Hanoi est incalculable.” Many variations were indeed studied and some of them we already encountered in previous chapters: the Linear TH in Chapter 2, problems in Chapter 3 allowing for irregular states, the Switching TH in Chapter 4, and the tasks in Chapter 5 where more than three pegs are available.

The Chinese Rings

In this chapter, we will discuss the mathematical theory of the CR which goes back to the booklet by Gros of 1872 [168]. This mathematical model may serve as a prototype for the approach to analyze other puzzles in later chapters. In Section 1.1 we develop the theory based on binary coding leading to a remarkable sequence to be discussed in Section 1.2. Some applications will be presented in Section 1.3.

Tower of Hanoi Variants with Oriented Disc Moves

In Chapter 6 we have introduced the concept of a TH variant and presented several such puzzles: the BWTH, additional variants with colored discs, and the BTH. We continued in Chapter 7 where the TL (and its variations) were treated in detail. In this chapter we turn our attention to the TH with oriented disc moves.