Asymptotic aspects of Schreier graphs and Hanoi Towers groups

We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior.Comment: 5 page

A new lower bound for the Towers of Hanoi problem

More than a century after its proposal, the Towers of Hanoi puzzle with 4 pegs was solved by Thierry Bousch in a breakthrough paper in 2014. The general problem with p pegs is still open, with the best lower bound on the minimum number of moves due to Chen and Shen. We use some of Bousch's new ideas to obtain an asymptotic improvement on this bound for all p >= 5

Student Monks – Teaching Recursion in an IS or CS Programming Course Using the Towers of Hanoi

Educators have been using the Towers of Hanoi problem for many years as an example of a problem that has a very elegant recursive solution. However, the elegance and conciseness of this solution can make it difficult for students to understand the amount of computer time required in the execution of this solution. And, like many recursive computer programs, students often find it difficult to follow a trace of the solution. Research in computer education has shown that active learning exercises achieve positive educational results. In line with this research, an active learning exercise was employed in the classroom to assist students in gaining a better understanding of the recursive solution to the Towers of Hanoi problem. This demonstration can be used in an introductory IS or CS programming class, independent of the language used. The demonstration involves using student volunteers, who, in the demonstration, are referred to as “monks,” a reference to the original problem that had monks moving the golden rings in the Towers of Hanoi. An anonymous student survey revealed that students felt strongly that the demonstration helped them better understand recursion, and that the demonstration was a good use of class time. In addition, an analysis of a small sample of students’ computer programs following the demonstration, suggests that there may be pedagogical benefits to use of the student monk demonstration

On a Generalization of the Hanoi Towers Group

In 2012, Bartholdi, Siegenthaler, and Zalesskii computed the rigid kernel for the only known group for which it is non-trivial, theHanoi towers group. There they determined the kernel was the Klein 4 group. We present a simpler proof of this theorem. In thecourse of the proof, we also compute the rigid stabilizers and present proofs that this group is a self-similar, self-replicating, regular branch group. We then construct a family of groups which generalize the Hanoi towers group and study the congruence subgroup problem for the groups in this family. We show that unlike the Hanoi towers group, the groups in this generalization are just infinite and have trivial rigid kernel. We also put strict bounds on the branch kernel. Additionally, we show that these groups have subgroups of finite index with non-trivial rigid kernel, adding infinitely many new examples. Finally, we show that the topological closures of these groups have Hausdorff dimension arbitrarily close to 1

PROLOG.

A Description is Given of Prolog, a Contraction of Programming in Logic, Which Uses the Formalism of Mathematical Logic as its Primary Design Principle. the Structure of Prolog is Examined, and a Database Program is Described to Illustrate its application. an Application to an Artificial Intelligence Problem, the Towers of Hanoi, is Also Given