## Tower of HanoiThe ## Rules of Tower of Hanoi:## 1. The Three Pegs: Source, Destination, and Spare:At the heart of the , and A, B. Each peg can hold the C, and your goal is to move all the disks from the source peg to the destination peg while using the disks as a temporary holding area.third peg## 2. Disk Sizes and Ordering:The ## 3. The Recursive Algorithm:The core strategy for solving the . Here's the recursive algorithm:'n'- Move the top
from the source peg to the spare peg using the destination peg as an intermediate step.*'n-1' disks* - Move the
from the source peg to the destination peg.*largest disk ('n')* - Move the
from the spare peg to the destination peg using the source peg as an intermediate step.*'n-1' disks*
## 4. Understanding Recursion:
, the algorithm keeps applying itself to the smaller sub-problems until it reaches the base case, which is moving a single disk and a task that's easy to achieve.Tower of Hanoi## 5. Solving the Puzzle:Start with the ## 6. The Beauty of Exponential Growth:One intriguing aspect of the Tower of Hanoi puzzle is the , the number of moves needed is 'n' disks. This rapid growth showcases the complex interplay between simple rules and complex outcomes.2^n - 1## 7. Patience and Strategy:While the rules of the Tower of Honzi puzzle are straightforward, successfully completing it requires patience and strategic thinking. You must carefully plan your moves and anticipate the outcomes of each step. The puzzle encourages you to explore various pathways and evaluate the consequences of your decisions. The puzzle may sound complex, but we can break it down into simple steps:
Think of the smallest version of this problem: just
Now, imagine you have and 'A', stacked on the source peg. To move these disks to the destination peg, you must follow these steps:'B'- Move disk
from the source peg to the spare peg.*'A'* - Move disk
from the source peg to the destination peg.*'B'* - Move disk
from the spare peg to the destination peg.*'A'*
As the number of disks increases, the puzzle becomes more interesting. Let's take three disks: . Here's how you solve it:'C'- First, move the top
to the*two disks ('A' and 'B')*, using the destination peg as an intermediate step. It follows the same steps we used for*spare peg*.*two disks* - Now, move the
to the destination peg.*largest disk ('C')* - Finally, move the two smaller disks (
and*'A'*) from the spare peg to the destination peg using the source peg as an intermediate step.*'B'*
The pattern continues as you add (four disks, and 'A', 'B', 'C''D'), the process looks like this:- Move the top three disks (
and*'A', 'B',*) to the spare peg using the destination peg as an intermediate step.*'C'* - Move the
to the destination peg.*largest disk ('D')* - Move the three smaller disks (
and*'A', 'B',*) from the spare peg to the destination peg using the source peg as an intermediate step.*'C'*
## Example:Let's take a program to illustrate the tower of Hanoi problem having 3 pegs:
Suppose you have a stack of disks and you want to move from , it's simple; you just move it directly. But when you have more than one disk, the recursive magic happens.one diskIf you want to move from the source peg to the spare peg (using the destination peg as temporary storage). It frees up the bottom disk, which you move directly to the destination peg. After that, you move the n-1 disks from the spare peg to the destination peg, using the source peg as temporary storage. It's like solving a smaller version of the same problem twice, once to clear the path for the largest disk and once to move the rest to the destination. This process keeps repeating for each smaller n-1 disks until you reach the base case of moving a sub-problem, which is straightforward. The beauty of recursion is that each sub-problem is solved using the same approach, making the solution elegant and easy to understand.single disk
## Example:Let's take a program to illustrate the
In the traditional (let's call them three pegs and A, B,), and the goal is to move a stack of disks from C to peg A, obeying certain rules. Extending this to peg C introduces additional complexity.five pegsIn a five-peg - You can only move
at a time.*one disk* - A larger disk cannot be placed on top of a smaller disk.
- You have
instead of three, and you need to move the entire stack from one peg*five pegs*to another peg*(source)*using the remaining pegs as temporary storage.*(destination)*
You can use a similar recursive approach to solve this extended problem, but it becomes more intricate. Here's a high-level explanation of the approach:
. If you want to move sub-problems from the source peg to the destination peg using five pegs, you can do the following:n disks- Move the
to one of the auxiliary pegs.*top n-2 disks* - Move the
to another auxiliary peg.*(n-1)-th disk* - Move the
to the destination peg.*n-th disk* - Move the
from the auxiliary peg to the destination peg.*(n-1)-th disk* - Move the
from the first auxiliary peg to the destination peg.*n-2 disks*
The key challenge in extending the and ensuring that the rules of the problem are followed. The auxiliaries helps to manage these complexities by solving smaller instances of the problem and combining their solutions to move the entire stack of disks.recursive approachNext TopicVigenere Cypher Program in C |