Tower Of Hanoi

1. The Three Pegs: Source, Destination, and Spare:

At the core of the Tower of Hanoi challenge lie three pegs, typically denoted as A, B, and C. Every peg has the capacity to accommodate disks, and the objective is to transfer all the disks from the initial peg to the target peg by leveraging the third peg as an intermediary storage space.

2. Disk Sizes and Ordering:

The disks are available in various sizes, arranged with the largest disk at the bottom and the smallest disk at the top. This specific configuration is crucial as it establishes the basis for the puzzle's difficulty. It's imperative to consistently maintain the rule of never placing a larger disk on top of a smaller one throughout the entire procedure.

3. The Recursive Algorithm:

The core strategy for solving the Tower of Hanoi puzzle revolves around a simple yet powerful pattern that scales with the number of disks, 'n' . Here's the recursive algorithm:

• ✅ Move the top 'n-1' disks from the source peg to the spare peg using the destination peg as an intermediate step.
• ✅ Move the largest disk ('n') from the source peg to the destination peg.
• ✅ Move the 'n-1' disks from the spare peg to the destination peg using the source peg as an intermediate step.

4. Understanding Recursion:

Recursion serves as a cornerstone principle in the realms of computer science and mathematics. This method revolves around decomposing a convoluted issue into more manageable, akin sub-issues. Within the Tower of Hanoi scenario, the algorithm continuously iterates over these reduced sub-problems until it hits the fundamental scenario of relocating a solitary disk, a task that is comparatively straightforward to accomplish.

5. Solving the Puzzle:

Commence with the original arrangement of disks on the starting peg to tackle the Tower of Hanoi challenge. Implement the recursive procedure to shift the disks from the source peg to the target peg while strictly following the specified regulations. While resolving the smaller sub-problems, the puzzle will progressively evolve into a series of more compact and solvable assignments.

6. The Beauty of Exponential Growth:

One fascinating feature of the Tower of Hanoi problem is the rapid increase in the number of steps needed when the quantity of disks rises. When 'n' disks are present, the total moves required amount to 2^n - 1. This swift expansion highlights the intricate relationship between basic regulations and intricate results.

7. Patience and Strategy:

Although the guidelines for solving the Tower of Honzi puzzle are clear, mastering it demands a combination of patience and strategic analysis. It is essential to meticulously strategize your actions and foresee the results of each move. This puzzle prompts you to investigate different approaches and assess the impacts of your choices.

The puzzle might appear intricate, yet we can simplify it by dividing it into straightforward stages:

1. Commence with a Singular Disk:

Think about the most basic scenario: a single disk. Shifting a single disk is a straightforward task. You elevate it from the original peg and position it onto the target peg.

1. When dealing with two disks:

Now, imagine you have two disks , labeled 'A' and 'B' , stacked on the source peg. To move these disks to the destination peg, you must follow these steps:

• ✅ Move disk 'A' from the source peg to the spare peg.
• ✅ Move disk 'B' from the source peg to the destination peg.
• ✅ Move disk 'A' from the spare peg to the destination peg.

1. Three Disks:

As the number of disks increases, the puzzle becomes more interesting. Let's take three disks: 'A', 'B' , and 'C' . Here's how you solve it:

• ✅ First, move the top two disks ('A' and 'B') to the spare peg , using the destination peg as an intermediate step. It follows the same steps we used for two disks .
• ✅ Now, move the largest disk ('C') to the destination peg.
• ✅ Finally, move the two smaller disks ( 'A' and 'B' ) from the spare peg to the destination peg using the source peg as an intermediate step.

1. Four Disks:

The pattern continues as you add more disks . For four disks ( 'A', 'B', 'C' , and 'D' ), the process looks like this:

• ✅ Move the top three disks ( 'A', 'B', and 'C' ) to the spare peg using the destination peg as an intermediate step.
• ✅ Move the largest disk ('D') to the destination peg.
• ✅ Move the three smaller disks ( 'A', 'B', and 'C' ) from the spare peg to the destination peg using the source peg as an intermediate step.

Example:

Let's consider a program that demonstrates the Tower of Hanoi issue with a total of 3 pegs:

#include <stdio.h>
void towerOfHanoi(int n, char source, char auxiliary, char destination) {
 if (n == 1) {
printf("Move disk 1 from %c to %c\n", source, destination);
 return;
 }
towerOfHanoi(n - 1, source, destination, auxiliary);
printf("Move disk %d from %c to %c\n", n, source, destination);
towerOfHanoi(n - 1, auxiliary, source, destination);
}
int main() {
 int n;
printf("Enter the number of disks: ");
scanf("%d", &n);
towerOfHanoi(n, 'A', 'B', 'C');
 return 0;
}

Output:

Move disk 1 from %c to %c
Move disk [value] from %c to %c
Enter the number of disks:

Explanation:

Working of Recursion:

Suppose you possess a collection of disks and aim to transfer them from one peg to another, leveraging a spare peg for interim storage. When dealing with a solitary disk, the process is straightforward as you can shift it directly. However, when managing multiple disks, this is where the recursive algorithm comes into play.

When tasked with relocating n disks, the process involves initially shifting the upper n-1 disks from the original peg to an additional peg (utilizing the final peg for interim storage). This action allows for the bottom disk to be accessible for direct transfer to the target peg. Subsequently, the n-1 disks are transferred from the extra peg to the target peg, with the original peg serving as temporary storage. This methodology essentially entails resolving a scaled-down version of the task twice: once to facilitate the movement of the largest disk and again to transition the remaining disks to the destination. This iterative pattern continues for each subsequent minor sub-task until reaching the fundamental scenario of relocating a solitary disk, which is a relatively simple action. The elegance of recursion lies in its consistent application across all sub-tasks, resulting in a refined and comprehensible solution.

Variables in the program:

n: The number of disks to be moved.

The origin peg from which disks are shifted.

The auxiliary pin is employed for temporary holding.

The target peg where disks are transferred.

Example:

Let's consider a program demonstrating the Tower of Hanoi puzzle with 5 rods:

#include <stdio.h>
void towerOfHanoi(int n, char source, char auxiliary1, char auxiliary2, char auxiliary3, char destination) {
 if (n == 1) {
printf("Move disk 1 from %c to %c\n", source, destination);
 return;
 }

 // Move n-2 disks to auxiliary peg 1
towerOfHanoi(n - 2, source, auxiliary2, auxiliary3, destination, auxiliary1);

 // Move disk n-1 to auxiliary peg 2
printf("Move disk %d from %c to %c\n", n - 1, source, auxiliary2);

 // Move disk n to destination peg
printf("Move disk %d from %c to %c\n", n, source, destination);

 // Move disk n-1 from auxiliary peg 2 to destination peg
printf("Move disk %d from %c to %c\n", n - 1, auxiliary2, destination);

 // Move n-2 disks from auxiliary peg 1 to destination peg
towerOfHanoi(n - 2, auxiliary1, source, auxiliary2, destination, auxiliary3);
}

int main() {
 int n;
printf("Enter the number of disks: ");
scanf("%d", &n);

 // Assuming you have five pegs: A, B, C, D, and E
towerOfHanoi(n, 'A', 'B', 'C', 'D', 'E');

 return 0;
}

Output:

Move disk 1 from %c to %c
Move disk [value] from %c to %c
Enter the number of disks:

Explanation:

In the classic Tower of Hanoi puzzle, there are three pegs denoted as A, B, and C. The objective is to transfer a series of disks from peg A to peg C while adhering to specific regulations. When expanding this to involve five pegs, it introduces further intricacies.

In a five-peg Tower of Hanoi problem, you still have a set of rules to follow:

• ✅ You can only move one disk at a time.
• ✅ A larger disk cannot be placed on top of a smaller disk.
• ✅ You have five pegs instead of three, and you need to move the entire stack from one peg (source) to another peg (destination) using the remaining pegs as temporary storage.

You have the option to employ a comparable recursive strategy to address this expanded issue, albeit with increased complexity. Below is a broad overview of the methodology:

Base Scenario: In the initial case, if there is only a single disk, you are able to transfer it directly from the starting peg to the target peg.

Recursive Step: When you have more than one disk, you need to break down the problem into smaller sub-problems . If you want to move n disks from the source peg to the destination peg using five pegs, you can do the following:

• ✅ Move the top n-2 disks to one of the auxiliary pegs.
• ✅ Move the (n-1)-th disk to another auxiliary peg.
• ✅ Move the n-th disk to the destination peg.
• ✅ Move the (n-1)-th disk from the auxiliary peg to the destination peg.
• ✅ Move the n-2 disks from the first auxiliary peg to the destination peg.

Recursion: Utilize the identical approach recursively to the subtasks. This procedure persists until you encounter the base scenario of relocating a solitary disk.

The primary difficulty in expanding the Tower of Hanoi puzzle to five pegs lies in the management of auxiliary peg selection and adherence to problem constraints. Leveraging the recursive method aids in addressing these intricacies by tackling smaller problem instances and merging their resolutions to shift the complete set of disks.