A Note on the Tower of Hanoi Problem with Evildoers

2021 ◽  
Vol 9 (1) ◽  
pp. 9-16
Author(s):  
S. M. Shahidul Islam ◽  
A. A. K. Majumdar
Keyword(s):  
Positive Integer ◽  
Explicit Form ◽  
Recent Literature ◽  
Tower Of Hanoi ◽  
New Variant ◽  
Minimum Number

Recent literature considers the variant of the classical Tower of Hanoi problem with n (³ 1) discs, where r (1 £ r < n) discs are evildoers, each of which can be placed directly on top of a smaller disc any number of times. Letting E(n, r) be the minimum number of moves required to solve the new variant, an explicit form of E(n, r) is available which depends on a positive integer constant N. This study investigates the properties of N.  

scholarly journals The Tower of Hanoi Problem with Evildoer Discs

2020 ◽  
Vol 43 (2) ◽  
pp. 205-209
Author(s):  
AAK Majumdar
Keyword(s):  
Explicit Form ◽  
Optimality Equation ◽  
Tower Of Hanoi ◽  
New Variant ◽  
Minimum Number ◽  
Academy Of Sciences

This paper deals with a variant of the classical Tower of Hanoi problem with n ( ≥ 1) discs, of which r discs are evildoers, each of which can be placed directly on top of a smaller disc any number of times. Denoting by E(n, r) the minimum number of moves required to solve the new variant, is given a scheme find the optimality equation satisfied by E(n, r). An explicit form of E(n, r) is then obtained. Journal of Bangladesh Academy of Sciences, Vol. 43, No. 2, 205-209, 2019

scholarly journals New variants of the bottleneck tower of Hanoi problems

2021 ◽  
Vol 44 (2) ◽  
pp. 197-200
Author(s):  
Abdullah Al Kafi Majumdar
Keyword(s):  
Closed Form ◽  
Explicit Form ◽  
Closed Form Expression ◽  
Form Expression ◽  
Tower Of Hanoi ◽  
New Variant ◽  
Minimum Number ◽  
Academy Of Sciences

This paper considers two variants of the bottleneck Tower of Hanoi problems with n (≥1) discs and the bottleneck size b (≥2), which allows violation of the “divine rule” (at most) once. Denoting by MB3(n, b) the minimum number of moves required to solve the new variant of the bottleneck Tower of Hanoi problem, an explicit form of MB3(n, b) is found. Also, MB4(n, b) denotes the minimum number of moves required to solve the new variant of the bottleneck Reve’s puzzle, a closed-form expression of MB4(n, b) is derived. Journal of Bangladesh Academy of Sciences, Vol. 44, No. 2, 197-200, 2020

scholarly journals The Reve’s Puzzle with Single Relaxation

2020 ◽  
Vol 43 (2) ◽  
pp. 211-212
Author(s):  
AAK Majumdar
Keyword(s):  
Explicit Form ◽  
New Variant ◽  
Minimum Number ◽  
Academy Of Sciences

This paper considers a variant of the Reve’s puzzle with n ( ≥ 1) discs which allows at most one violation of the “divine rule”. Denoting by S4(n) the minimum number of moves required to solve the new variant, an explicit form of S4(n) is given. Journal of Bangladesh Academy of Sciences, Vol. 43, No. 2, 211-212, 2019

scholarly journals The Reve’s Puzzle with an Evildoer Disc

2020 ◽  
Vol 44 (1) ◽  
pp. 53-58
Author(s):  
AAK Majumdar
Keyword(s):  
Explicit Form ◽  
Optimality Equation ◽  
New Variant ◽  
Minimum Number ◽  
Academy Of Sciences

This paper considers a variant of the Reve’s puzzle with n (≥1) discs and an evildoer, which can be placed directly on top of a smaller disc any number of times. Denoting by E(n) the minimum number of moves required to solve the new variant, we give a scheme to find the optimality equation satisfied by E(n). We then find an explicit form of E(n). Journal of Bangladesh Academy of Sciences, Vol. 44, No. 1, 53-58, 2020

scholarly journals On the Star Puzzle

2019 ◽  
Vol 39 ◽  
pp. 1-14
Author(s):  
AAK Majumdar
Keyword(s):  
Recurrence Relation ◽  
Additional Condition ◽  
Tower Of Hanoi ◽  
Minimum Number

In the star puzzle, there are four pegs, the usual three pegs, S, P and D, and a fourth one at 0. Starting with a tower of n discs on the peg P, the objective is to transfer it to the peg D, in minimum number of moves, under the conditions of the classical Tower of Hanoi problem and the additional condition that all disc movements are either to or from the fourth peg. Denoting by MS(n) the minimum number of moves required to solve this variant, MS(n) satisfies the recurrence relation . This paper studies rigorously and extensively the above recurrence relation, and gives a solution of it. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 1-14

scholarly journals Memory in a New Variant of King’s Family for Solving Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1251
Author(s):  
Munish Kansal ◽  
Alicia Cordero ◽  
Sonia Bhalla ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Convergence Analysis ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Recent Literature ◽  
Divided Difference ◽  
Difference Operators ◽  
New Variant ◽  
Order Four ◽  
With Memory

In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King’s family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.

On the metric dimension and diameter of circulant graphs with three jumps

2018 ◽  
Vol 10 (01) ◽  
pp. 1850008
Author(s):  
Muhammad Imran ◽  
A. Q. Baig ◽  
Saima Rashid ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Metric Spaces ◽  
Metric Dimension ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Infinite Class ◽  
Metric Properties ◽  
Minimum Number ◽  
Metric Basis

Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.

scholarly journals Region Coloring in Minahasa Regency Using Sequential Color Algorithm

d'CARTESIAN ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 86
Author(s):  
Yevie Ingamita ◽  
Nelson Nainggolan ◽  
Benny Pinontoan
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Chromatic Number ◽  
Human Life ◽  
Minimum Number ◽  
Mathematical Sciences ◽  
Map Coloring ◽  
Graph Application

Graph Theory is one of the mathematical sciences whose application is very wide in human life. One of theory graph application is Map Coloring. This research discusses how to color the map of Minahasa Regency by using the minimum color that possible. The algorithm used to determine the minimum color in coloring the region of Minahasa Regency that is Sequential Color Algorithm. The Sequential Color Algorithm is an algorithm used in coloring a graph with k-color, where k is a positive integer. Based on the results of this research was found that the Sequential Color Algorithm can be used to color the map of Minahasa Regency with the minimum number of colors or chromatic number χ(G) obtained in the coloring of 25 sub-districts on the map of Minahasa Regency are 3 colors (χ(G) = 3).

scholarly journals On a Decomposition Method in the Problem of Operation Speed for a Linear Discrete System with Bounded Control

2019 ◽  
Vol 09 (4) ◽  
pp. 157-161
Author(s):  
D.N. Ibragimov ◽  
E.E. Turchak
Keyword(s):  
Explicit Form ◽  
Discrete System ◽  
Decomposition Theorem ◽  
Two Dimensional ◽  
State Matrix ◽  
Bounded Control ◽  
Operation Speed ◽  
Linear Discrete System ◽  
Minimum Number ◽  
Main Decomposition

The article presents the problem of operation speed for a linear discrete system with bounded control. For the case when the minimum number of steps necessary for the system to reach zero significantly exceeds the dimension of the phase space, a method of decomposition into scalar and two-dimensional subsystems is developed, based on the reduction of the state matrix to normal Jordan form. Moreover, due to the developed algorithm for adding two polyhedrons with linear complexity, it is possible to construct sets of 0-controllability for two-dimensional subsystems in an explicit form. A description of the main tools for solving the problem of operation speed is also presented, as well as the statement of the decomposition problem. Further, some properties of polyhedrons in the plane are formulated and proved, on the basis of which an algorithm for calculating the set of vertices of the sum of two polyhedrons in R2 in explicit form is developed. In conclusion, the main decomposition theorem is formulated and proved. And on the basis of the developed methods, the solution to the problem of the optimal damping speed of a high-rise structure located in the zone of seismic activity was constructed.

scholarly journals Logicamente: uma Plataforma Digital para Estimulação das Funções Executivas

2021 ◽  
Vol 22 (4) ◽  
pp. 488-492
Author(s):  
Anderson Sampaio Carapiá ◽  
Larissa Ribeiro Bastos ◽  
Márcio Renê Brandão Soussa ◽  
Lynn Rosalina Gama Alves ◽  
William de Souza Santos
Keyword(s):  
Academic Performance ◽  
Executive Functions ◽  
Adult Female ◽  
Learning Process ◽  
Age Groups ◽  
Digital Games ◽  
Minimum Amount ◽  
Tower Of Hanoi ◽  
Minimum Number ◽  
Game Platform

Resumo As funções executivas têm um papel fundamental para o desempenho acadêmico, pois elas são responsáveis pelo controle, monitoramento e regulação das ações, pensamentos e emoções. Desta forma, o estímulo a essas funções pode trazer contribuições importantes para o processo de aprendizagem e uma das estratégias que vem sendo utilizada para essa estimulação é através do uso de variados jogos digitais. Portanto, este artigo tem por objetivo relatar os achados iniciais das interações de diversos usuários com o jogo Torre de Hanói, que compõe a plataforma de jogos intitulada LogicaMente. Para tanto, foram coletadas informações de 3.524 partidas realizadas por 1.324 jogadores de diferentes faixas etárias e de ambos os sexos e dentre os resultados, foi identificado que apenas 32,5% dos jogadores conseguiram finalizar o jogo Torre de Hanói com o número mínimo de movimentações no modo fácil (com 3 discos) e menos ainda, 12,2% no modo difícil (com 4 discos). Como conclusão, foi identificado que houve um predomínio do público adulto de gênero feminino, que os interatores demonstram ter uma dificuldade em concluir o jogo na menor quantidade de movimentos, mas observou-se que a persistência pode ter sido um dos motivadores para que os jogadores obtivessem êxito a partir das suas novas tentativas. Palavras-chave: Funções Executivas. Jogos Digitais. Torre de Hanói. Abstract The executive functions play a key role in academic performance, as they are responsible for controlling, monitoring, and regulating actions, thoughts, and emotions. In this way, stimulating these functions can bring contributions for the learning process and one of the approaches that have been used for this stimulation is through the use of various digital games. Therefore, this article aims at reporting initial findings of the interactions of several users with the game Tower of Hanoi, which is part of the game platform entitled LogicaMente. For this purpose, information were collected over 3,524 matches played by 1,324 players from different age groups and both genders and among the results, it was identified that only 32.5% of the players managed to finalize the Tower of Hanoi game with the minimum number of movements in easy mode (with 3 discs) and even less, 12.2% in difficult mode (with 4 discs). As a conclusion, it was noticed a predominance of adult female players, the interactors had difficulty in completing the game with the minimum amount of movements. However, it was observed that persistence may have been one of the motivators for the players to succeed from their new attempts. Keywords: Executive Functions. Digital games. Tower of Hanoi.

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