What is the least number of moves needed to solve the k-peg Towers of Hanoi problem?

We prove that the solutions to the [Formula: see text]-peg Tower of Hanoi problem given by Frame and Stewart are minimal. The proof relies on first identifying that for any [Formula: see text]-disk, [Formula: see text]-peg problem, there is at least one minimal sequence is symmetric. We show that if we order the number moved required for the disks in the minimal symmetric sequence in an increasing manner and obtain the sequence [Formula: see text], then [Formula: see text]. We also prove that the maximum number of disks that can be moved using [Formula: see text] steps is [Formula: see text]. We use these to lower bound the telescopic sum (2) that is a lower bound on the number of moves required for any minimal symmetric sequence. This gives us the required result.

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 $p \geq 5$.

The Application of Hanoi Towers Game in Logistics Management

In this chapter we discussed the application of Towers of Hanoi in logistics management applications, for this purpose. Firstly, we discussed how pile problems applications impact in logistics, later we discussed the Hanoi Towers application in Logistics Management, its mathematical model and common solutions applying different paradigms (iterative, recursive, and heuristics), we present an in depth analysis in genetic algorithms, later we present the analysis of a genetic algorithm applied to solve basic Hanoi Tower game (three pegs three discs) implemented in C language, and later we discuss its generalization from three to four discs. Finally, we discussed preliminary results and present our conclusions. Main contribution of this chapter is demonstrate game theory, specifically Towers of Hanoi, can be applied to solve logistics problems, and an approach for the generalization of basic Hanoi Tower form, three discs to four discs, by means of genetic algorithms implemented in C language.

The Mathematics of Various Entertaining Subjects

The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books exploring puzzles and brainteasers, research in recreational mathematics has often been neglected. This book brings together authors from a variety of specialties to present fascinating problems and solutions in recreational mathematics. The chapters show how sophisticated mathematics can help construct mazes that look like famous people, how the analysis of crossword puzzles has much in common with understanding epidemics, and how the theory of electrical circuits is useful in understanding the classic Towers of Hanoi puzzle. The card game SET® is related to the theory of error-correcting codes, and simple tic-tac-toe takes on a new life when played on an affine plane. Inspirations for the book's wealth of problems include board games, card tricks, fake coins, flexagons, pencil puzzles, poker, and so much more. Looking at a plethora of eclectic games and puzzles, this book is sure to entertain, challenge, and inspire academic mathematicians and avid math enthusiasts alike.

Dimer–monomer model on the Towers of Hanoi graphs

The number of dimer–monomers (matchings) of a graph [Formula: see text] is an important graph parameter in statistical physics. Following recent research, we study the asymptotic behavior of the number of dimer–monomers [Formula: see text] on the Towers of Hanoi graphs and another variation of the Sierpiński graphs which is similar to the Towers of Hanoi graphs, and derive the recursion relations for the numbers of dimer–monomers. Upper and lower bounds for the entropy per site, defined as [Formula: see text], where [Formula: see text] is the number of vertices in a graph [Formula: see text], on these Sierpiński graphs are derived in terms of the numbers at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accuracy.

Intelligent System Design Using Hyper-Heuristics

Determining the most appropriate search method or artificial intelligence technique to solve a problem is not always evident and usually requires implementation of the different approaches to ascertain this. In some instances a single approach may not be sufficient and hybridization of methods may be needed to find a solution. This process can be time consuming. The paper proposes the use of hyper-heuristics as a means of identifying which method or combination of approaches is needed to solve a problem. The research presented forms part of a larger initiative aimed at using hyper-heuristics to develop intelligent hybrid systems. As an initial step in this direction, this paper investigates this for classical artificial intelligence uninformed and informed search methods, namely depth first search, breadth first search, best first search, hill-climbing and the A* algorithm. The hyper-heuristic determines the search or combination of searches to use to solve the problem. An evolutionary algorithm hyper-heuristic is implemented for this purpose and its performance is evaluated in solving the 8-Puzzle, Towers of Hanoi and Blocks World problems. The hyper-heuristic employs a generational evolutionary algorithm which iteratively refines an initial population using tournament selection to select parents, which the mutation and crossover operators are applied to for regeneration. The hyper-heuristic was able to identify a search or combination of searches to produce solutions for the twenty 8-Puzzle, five Towers of Hanoi and five Blocks World problems. Furthermore, admissible solutions were produced for all problem instances.