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Hanoi graphs H model the Tower of Hanoi game with p pegs and n discs. Sierpinski graphs S arose in investigations of universal topological spaces and have meanwhile been studied extensively. It is proved that S embeds as a spanning subgraph into H if and only if p is odd or, trivially, if n = 1.

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 (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.

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.

In Chapter 2 we have described the fundamental object of the book—the classical TH with three pegs. We have revealed its secrets and hopefully convinced the reader that it contains exciting problems with not too difficult solutions. Now, a well-known mathematical metatheorem asserts that it is easy to generalize. In this chapter we are going to study the most natural and obvious generalization of the TH with three pegs, namely the TH with more than three pegs. The main message of the chapter is that in this particular case the abovementioned metatheorem is as wrong as it can possibly be. In order to avoid confusion with the introduction of a second parameter besides the number n of discs, namely the number p of pegs, we will focus on p = 4 in the first part of this chapter. Later we will extend our discussion to the general case.

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.

With the theory developed in Chapter 2 it was not very difficult to determine the age of the universe, see Exercise 2.8. Similarly, the time when the end of the universe is to be expected can be computed. The situation with the end of time is more delicate though, especially now that The Reve’s Puzzle has been solved and the solution is readily available in the present edition of our book. It cannot be excluded that the Brahmins will be exposed to the book and consequently come to the idea to save some time and use one more peg—the Devil’s one. Luckily for us, the Brahmins are men of great honor and will therefore not deviate from “the fixed and immutable laws” imposed by the Trimurti of creation, maintenance, and destruction of the world represented as the three “diamond needles” corresponding to the triad of dieties Brahma, Vishnu, and Shiva. So we are safe for some time until we might be faced with dramatic events in the history of the universe.

A numerical simulation of the elastic behaviour of the human mandible is realized via the finite element method. To test the validity of the simplifying assumptions of this model, carefully designed in vitro experiments are implemented on a special-purpose biomechanical test stand. Using this validation procedure, a method for identifying the material properties of the mandible can be implemented by solving an inverse problem. Resume. Une simulation numerique du comportementde la mandibule humaine ae realiseel'aide de la methode desements finis. Pour tester la validite des hypotheses simplifi- catrices du modele, des experiences soigneusement projetees ontete performees sur un banc d'essai biomecanique. Employant ce procede de validation, une methode d'identifier les proprietes materiels de la mandibule peutimplemente par la solution d'un probleme inverse.

Local and global regularity properties of weak solutions of the Schrödinger equation −Δu+qu=λu play an important role in the spectral theory of the corresponding operator [Formula: see text]. Central among these properties is local boundedness of the solutions u, which is derived in an elementary way for potentials q whose negative parts q− lie in the local Kato class K loc . The method also provides mean value inequalities for and, in case q+ is in K loc too, continuity of u. To employ these mean value inequalities for bounds on eigenfunctions of T in a fixed direction, classes Kρ are introduced which reflect the behavior of q at infinity. A couple of examples allow to compare these classes with more conservative ones like the Stummel class Q and the global Kato class K. The fundamental property of local boundedness of solutions also serves as a base for a very short proof of the self-adjointness of T if the operator is bounded from below and q−∈K loc . If q(x) is permitted to go to −∞, as |x|→∞, a large class K ρ which guarantees self-adjointness of T is derived and contains the case q−(x)= O (|x|2). The Spectral Theorem then allows to deduce rapidly decaying bounds on eigenfunctions for discrete eigenvalues, at least if q−(x)= o (|x|2). This is also the condition under which the existence of a bounded solution is sufficient to guarantee λ∈σ(T). Here q−(x)= O (|x|2) appears as a borderline case and is discussed at some length by means of an explicit example. The class of admissible operators extending to these borderline cases with potentials singular locally and at infinity, the regularity results for solutions being mostly optimal, as demonstrated by numerous examples, yet the proofs being shorter and more straightforward than those to be found in literature for smaller classes and weaker results, the sets Kρ under consideration and the methods employed appear to be quite natural.

The canonical distance of points on the Sierpinski gasket is considered and its expectation deduced. The solution is surprising, both for the value and for the method derived from an analysis of graphs connected with the Tower of Hanoi problem.

Ultrasound microscopy is a promising tool to investigate material properties of various tissues. The application to bone is not straightforward and requires a thorough inspection of currently accepted models. Their refinements will be incorporated into a computer program for an objective interpretation of the measured data, based on the numerical solution of inverse problems. Resume. La microscopie ultra-son est un outil prometteur pour l'exploration des proprietes de materiel de tissus varies. L'application au tissu osseux n'est pourtant pas facilrealiser et exige une inspection scrupuleuse des modeles acceptespresent. Leurs ameliorations seront incorporees dans un programme d'ordinateur pour une interpretation objective des dates mesures, fonde sur les solutions numeriques de problemes inverses.

(1992). Pascal's Triangle and the Tower of Hanoi. The American Mathematical Monthly: Vol. 99, No. 6, pp. 538-544.

Dans L 2 (R n ), n≥2, on etudie l'operateur −Δ+q defini sur Co ∞ (R n ) avec q∈L 2,loc (R n ) a valeurs reelles. On etablit le lien entre la decroissance des fonctions propres et la distance de λ au spectre essentiel

On determine le spectre essentiel des operateurs de Schrodinger avec un potentiel a symetrie spherique V(x)=q(|x|)

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 Stern’s 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.