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 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 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.

scholarly journals Generalized multi-peg tower of hanoi problem

1996 ◽  
Vol 38 (2) ◽  
pp. 201-208 ◽  
Author(s):  
A. A. K. Majumdar
Keyword(s):  
Dynamic Programming ◽  
Explicit Expression ◽  
Programming Technique ◽  
Optimality Equation ◽  
Tower Of Hanoi ◽  
Minimum Number ◽  
Explicit Expressions

AbstractThis paper treats the multi-peg generalization of the Tower of Hanoi problem with n(≥ 1) disks and p(≥ 3) pegs, P1, P2,…, Pp. Denoting by M(n, p) the presumed minimum number of moves required to transfer the tower of n disks from the source peg, P1, to the destination peg, Pp, under the condition that each move transfers the topmost disk from one peg to another such that no disk is ever placed on top of a smaller one, the Dynamic Programming technique has been employed to find the optimality equation satisfied by M(n, p). Though an explicit expression for M(n, p) is given, no explicit expressions for the partition numbers (at which M(n, p) is attained) are available in the literature for p ≥ 5. The values of the partition numbers have been given in this paper.

scholarly journals A Note on the Frame-Stewart Conjecture on the Generalized Tower of Hanoi Problem

2019 ◽  
Vol 43 (1) ◽  
pp. 79-83
Author(s):  
AAK Majumdar
Keyword(s):  
Lower Bound ◽  
Tower Of Hanoi ◽  
Minimum Number ◽  
Academy Of Sciences

The generalized Tower of Hanoi with p (≥ 3) pegs and n (≥ 1) discs, proposed by Stewart (1939) is well-known. To solve the problem, the scheme followed is : First, move the tower of the topmost i (smallest, consecutive) discs (optimally) to one of the auxiliary pegs in a tower, using the p pegs; next, move the remaining n – i (largest) discs (optimally) to the destination peg in a tower, using the p – 1 pegs available; and finally, transfer the discs on the auxiliary peg to the destination peg (optimally) in a tower. This is the so-called Frame-Stewart conjecture, which remains to be settled. The minimum number of moves under the scheme is denoted by SF(n, p). Chen and Shen (2004) have re-considered the Frame-Stewart conjecture in more detail, and claimed that SF(n, p) is of the order of 2 [ n ( p 2 )!] 1 / ( p 2 ). This paper gives a better lower bound of SF(n, p), which shows that the claim of Chen et al. (2004) is not correct. Journal of Bangladesh Academy of Sciences, Vol. 43, No. 1, 79-83, 2019

scholarly journals On a Recurrence Relation Related to The Reve’s Puzzle

2018 ◽  
Vol 42 (2) ◽  
pp. 191-199
Author(s):  
AAK Majumdar
Keyword(s):  
Recurrence Relation ◽  
Value Function ◽  
Optimal Value Function ◽  
Optimality Equation ◽  
Tower Of Hanoi ◽  
Optimal Value ◽  
Academy Of Sciences ◽  
Local Value

The 4-peg Tower of Hanoi problem, commonly known as the Reve’s puzzle, is well-known. Motivated by the optimality equation satisfied by the optimal value function M(n) satisfied in case of the Reve’s puzzle, (Matsuura et al. 2008) posed the following generalized recurrence relation T(n, a) = min {aT(n-t, a)+S(t,3)}             1≤ t ≤ n where n ≥ 1 and a ≥ 2 are integers, and S(t, 3) = 2t – 1 is the solution of the 3-peg Tower of Hanoi problem with t discs. Some local-value relationships are satisfied by T(n, a) (Majumdar et al. 2016). This paper studies the properties of  T(n+1, a) – T(n, a) more closely for the case when a is an integer not of the form 2i for any integer i ≥ 2. Journal of Bangladesh Academy of Sciences, Vol. 42, No. 2, 191-199, 2018

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 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 Several Bounds for the K-Tower of Hanoi Puzzle

2017 ◽  
Vol 4 (7) ◽  
pp. 79-118
Author(s):  
Stephen B. Gregg ◽  
Britney Hopkins ◽  
Kristi Karber ◽  
Thomas Milligan ◽  
Johnny Sharp
Keyword(s):  
Upper Bounds ◽  
Tower Of Hanoi ◽  
Special Cases ◽  
Minimum Number

Abstract We consider special cases of a modified version of the Tower of Hanoi puzzle and demonstrate how to find upper bounds on the minimum number of moves that it takes to complete these cases.

Information capacity and bandwidth limitations in the SEM

1989 ◽  
Vol 47 ◽  
pp. 84-85
Author(s):  
D. C. Joy ◽  
R. D. Bunn
Keyword(s):  
Signal To Noise Ratio ◽  
Critical Dimension ◽  
Information Capacity ◽  
Acquisition Time ◽  
Signal To Noise ◽  
Sem Image ◽  
Probe Size ◽  
Minimum Number ◽  
Noise Ratio ◽  
Signal Channel

The information available from an SEM image is limited both by the inherent signal to noise ratio that characterizes the image and as a result of the transformations that it may undergo as it is passed through the amplifying circuits of the instrument. In applications such as Critical Dimension Metrology it is necessary to be able to quantify these limitations in order to be able to assess the likely precision of any measurement made with the microscope.The information capacity of an SEM signal, defined as the minimum number of bits needed to encode the output signal, depends on the signal to noise ratio of the image - which in turn depends on the probe size and source brightness and acquisition time per pixel - and on the efficiency of the specimen in producing the signal that is being observed. A detailed analysis of the secondary electron case shows that the information capacity C (bits/pixel) of the SEM signal channel could be written as :

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