Explain Tower of Hanoi with all necessary steps by considering basic rules.

I. Introduction

The Tower of Hanoi, also known as the Tower of Brahma or Lucas' Tower, is a mathematical game or puzzle. It consists of three rods and a number of disks of different sizes, which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.

The game follows three rules:

1. Only one disk can be moved at a time.
2. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty rod.
3. No disk may be placed on top of a smaller disk.

II. Detailed Step-by-Step Solution

The Tower of Hanoi is a classic example of a problem that can be solved using recursion. The solution involves moving the disks between the rods following the rules mentioned above. The minimum number of moves required to solve a Tower of Hanoi puzzle is 2^n – 1, where n is the number of disks.

Here are the detailed steps to solve the problem for n disks:

1. Move the top n-1 disks from Source to Auxiliary peg, using Destination peg. This is done by recursively applying the Tower of Hanoi solution. If there is only one disk, this step is skipped.

2. Move the nth disk from Source to Destination peg. This is a simple move, as it involves moving the largest disk that is not obstructed by any other disk.

3. Move the n-1 disks from Auxiliary peg to Destination peg, using Source peg. Again, this is done by recursively applying the Tower of Hanoi solution. If there is only one disk, it is simply moved to the Destination peg.

III. Program to solve Tower of Hanoi

Here is a Python program that solves the Tower of Hanoi problem using recursion:

def TowerOfHanoi(n , source, destination, auxiliary):
   if n==1:
       print("Move disk 1 from source",source,"to destination",destination)
       return
   TowerOfHanoi(n-1, source, auxiliary, destination)
   print("Move disk",n,"from source",source,"to destination",destination)
   TowerOfHanoi(n-1, auxiliary, destination, source)

# Driver code
n = 3
TowerOfHanoi(n,'A','B','C') 

IV. Examples

Let's consider an example of solving the Tower of Hanoi problem for 3 disks. The rods are named A (Source), B (Destination), and C (Auxiliary).

1. Move disk 1 from A to C.
2. Move disk 2 from A to B.
3. Move disk 1 from C to B.
4. Move disk 3 from A to C.
5. Move disk 1 from B to A.
6. Move disk 2 from B to C.
7. Move disk 1 from A to C.

The above steps are the result of the execution of the Python program provided. Diagrams can be used to illustrate the movement of disks from one rod to another, but it is not necessary for understanding the solution.

V. Conclusion

The Tower of Hanoi is a classic problem that demonstrates the power of recursion in problem-solving. The solution involves a series of moves that follow a simple pattern, which can be easily implemented in a computer program. The time complexity of the solution is O(2^n), which means the number of moves grows exponentially with the number of disks. This makes the Tower of Hanoi a computationally intensive problem for a large number of disks.

Summary

The Tower of Hanoi is a mathematical game or puzzle that involves moving disks between rods following certain rules. The minimum number of moves required to solve the puzzle is 2^n – 1, where n is the number of disks. The solution can be implemented using recursion.

Analogy

The Tower of Hanoi can be compared to a game where you have to move a stack of different-sized plates from one peg to another, following specific rules. It requires careful planning and strategic thinking to solve the puzzle.