Tower Of Hanoi

Research Report

21 December 2015

Prepared by:

Faizan Ahmed

EP – 1449027

Submitted To:

Sir Usman Amjad

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Acknowledgements

To complete this assignment, we took the some experience and

guideline of some respected persons, who deserve their thank-full

attitude. The completion of this report gives me much Pleasure. I would

like to give my Respect to Mr. Usman Amjad, Course Instructor, 402

for giving us a good chance for this assignment although numerous

consultations. I would also like to pay my deepest respect to all those

who have directly and indirectly guided us in writing this report.

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Contents

Acknowledgements ............................................................................................................. i

Executive Summary ............................................................................................................ 1

1 Introduction .................................................................................................................. 2

1.1 Background............................................................................................................. 2

1.2 Objectives ............................................................................................................... 3

1.3 Scope ..................................................................................................................... 4

2 Recursive Pattern of Hanoi .......................................................................................... 6

3 Methodology ................................................................................................................. 9

3.1 Research Questions ............................................................................................... 9

3.2 Research Design .................................................................................................... 9

3.3 Sample ................................................................................................................. 10

4 Algorithm Explaination .............................................................................................. 11

5 C# Hanoi Application with Code: ............................................................................... 13

6 Results ........................................................................................................................ 18

6.1 Research Questions: ............................................................................................ 18

7 References .................................................................................................................. 18

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Executive Summary

Abstract

We are going to examine a variation of the famous “Tower of Hanoi puzzle” that is posed but not solved, in a 1944 paper by Scorer et al, In this variation, disks of adjacent sizes can be moved, by the provided position that they are at the top of their respective stacks. We present an algorithm for solving this problem, and analyze its performance, and prove that it is optimal.

Methodology

In this research, we are going to discuss Iterative and Recursive problem “Tower of Hanoi”.

Key Terms Used

Tower of Hanoi

Representation approach

Algorithms

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1 Introduction

1.1 Background

This puzzle was invented by the French mathematician Édouard Lucas in 1883. There

is a story about an Indian temple in Kashi Vishwanath which contains a large room

with three time-worn posts in it surrounded by 64 golden disks. Brahmin priests,

acting out the command of an ancient prophecy, have been moving these disks, in

accordance with the immutable rules of the Brahma, since that time. The puzzle is

therefore also known as the Tower of Brahma puzzle.

This link is for reference

https://en.wikipedia.org/wiki/Tower_of_Hanoi#cite_note-2

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1.2 Objectives

The objectives of the report are to:

1- Introduction

2- Review of Key terms

3- Algorithm Explanation for each method

4- Application

5- Complete Code

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1.3 Scope

The Tower of Hanoi has been studied extensively in both Artificial Intelligence and

Psychology.

The Tower of Hanoi is used to solve problems hierarchically.

Where 'Hierachy' defines arranged in order of rank.

For this Recursive pattern of Hanoi is to be used to resolve problems.

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2 Recursive Pattern of Hanoi

Since this is a classic puzzle. Our goal is to transfer all the pieces from the left position 1 to

the right position 3.

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Rule: We can only move one disk at a time and you can never put a bigger disk on top of a

smaller disk.

From the Algorithm to transfer one, two, and three disks, we can easily find a recursive

pattern for additional disks.

- a pattern that uses information from one step to find the next step

- for moving n disks from post A to post C:

First, transfer n-1 disks from post A to post B. The number of moves will be the same as

those needed to transfer n-1 disks from post A to post C. Call this number M moves. [As you

can see above, with three disks it takes 3 moves to transfer two disks (n-1) from post A to

post C.]

Next, transfer disk 1 to post C [1 move].

Finally, transfer the remaining n-1 disks from post B to post C. [Again, the number of moves

will be the same as those needed to transfer n-1 disks from post A to post C, or M moves.]

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Therefore the number of moves needed to transfer n disks from post A to post C is 2M+1,

where M is the number of moves needed to transfer n-1 disks from post A to post C.

However, the recursive pattern can help us generate more numbers to find an explicit (non-

recursive) pattern. Here's how to find the number of moves needed to transfer larger

numbers of disks from post A to post C, remembering that M = the number of moves needed

to transfer n-1 disks from post A to post C:

for 1 disk it takes 1 move to transfer 1 disk from post A to post C;

for 2 disks, it will take 3 moves: 2M + 1 = 2(1) + 1 = 3

for 3 disks, it will take 7 moves: 2M + 1 = 2(3) + 1 = 7

for 4 disks, it will take 15 moves: 2M + 1 = 2(7) + 1 = 15

for 5 disks, it will take 31 moves: 2M + 1 = 2(15) + 1 = 31

And simply:

The minimum number of moves required to solve a Tower of Hanoi puzzle is 2n - 1, where n

is the number of disks.

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3 Methodology

3.1 Research Questions

The research questions to be answered by this Report are:

1- Introduction

2- Review of Key terms

3- Algorithm Explanation for each method

4- Application

5- Complete Code

3.2 Research Design

This Report will use Descriptive methods to answer the research questions. The following

table summarises the questions and the other methods are used to answer the each

questions:

Table 1 Methods used to answer research questions

Research Question Method Used to Answer Question

Introduction Mathematical Puzzle

Algorithm Recursive

Minimum moves Define in this paper

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3.3 Sample

Some samples for this report are used that are:

https://www2.bc.edu/~grigsbyj/Spafford_Final.pdf

www.ro.uow.edu.au/cgi/viewcontent.cgi?article=1017&context=compsciwp

www.cs.wm.edu/~pkstoc/gov.pdf

www.cs.nyu.edu/courses/summer07/G22.2340-001/.../McCann.pdf

www.cs.rutgers.edu/~szegedy/PUBLICATIONS/tower1.pdf

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4 Algorithm Explaination

Explaination:

The algorithm is pretty clearly show that we first remove n − 1 discs to get access to the nth column. And then we have to move the discs first to another peg after than, where you want the full tower. The algorithm has three arguments every time, ignoring the number of discs: A source peg, a destination peg and a temporary peg on which discs are initially set, in between (where every disc with size n − 1 fits). The recursion happens actually there two times, in every time procedure call. Once before the WRITELN and once after that. The one before the WRITELN will move n − 1 discs onto the temporary peg, using the destination peg as temporary storage (the arguments in the recursive call are in different order). After that, the remaining disc will be moved to the destination peg and afterwards the second recursion completes the moving of the entire tower, by moving the n − 1 tower from the temp peg to the destination peg, above disc n.

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Here is Algorithm:

procedure Hanoi(n: integer; source, dest, by: char);

Begin

if (n=1) then

writeln('Move the plate from ', source, ' to ', dest)

else begin

Hanoi(n-1, source, by, dest);

writeln('Move the plate from ', source, ' to ', dest);

Hanoi(n-1, by, dest, source);

end;

End;

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5 C# Hanoi Application with Code:

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Complete Code:

using System.Collections.Generic; using System.Linq; namespace HanoiTower { public class HanoiTowerClass { private int steps = 0; Stack<int> Tower1 = new Stack<int>(); Stack<int> Tower2 = new Stack<int>(); Stack<int> Tower3 = new Stack<int>(); private void Reset() { Tower1.Clear(); Tower2.Clear(); Tower3.Clear(); } private Stack<int> GetTower(Towers tower) { if (tower == Towers.Left) return Tower1; else if (tower == Towers.Middle) return Tower2; else return Tower3; } public List<int> GetTowerLeft() { return Tower1.ToList<int>(); } public List<int> GetTowerMiddle() { return Tower2.ToList<int>(); } public List<int> GetTowerRight() { return Tower3.ToList<int>(); } public void Init(int steps) { Reset(); this.steps = steps > Constant.MaxSteps ? Constant.MaxSteps : steps; for (int i = this.steps; i > 0; i--) Tower1.Push(i); }

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public int GetStep(Towers from) { Stack<int> fromTower = GetTower(from); if (fromTower.Count < 1) return 0; return fromTower.Peek(); } public bool Move(Towers from, Towers to) { int fromStep = 0; int toStep = 0; Stack<int> fromTower = GetTower(from); ; Stack<int> toTower = GetTower(to); ; if (fromTower.Count < 1) return false; fromStep = fromTower.Peek(); if (toTower.Count < 1) toStep = 10; else toStep = toTower.Peek(); if (fromStep < toStep) { toTower.Push(fromTower.Pop()); return true; } else if (fromStep == toStep) return true; else return false; } public bool IsComplete() { return Tower1.Count == 0 && (Tower2.Count == steps || Tower3.Count == steps); } } public enum Towers { Left, Middle, Right } public class Constant { public const int BottomTop = 333; public const int StepThickness = 25; public const int ColumnWidth = 150; public const int HandTop = 20; public const int Offset = 12; public const int MaxSteps = 6; } }

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Screenshots:

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6 Results

6.1 Research Questions:

Hance all research questions are answered.

7 References

All necessary references are provided in related sections.

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