Question

The puzzle called the Towers of Hanoi consists of three pegs, one of which contains several rings stacked in order of descending diameter from bottom to top. The problem is to move the stack of rings to another peg. You are allowed to move only one ring at a time, and at no time is a ring to be placed on top of a smaller one. Observe that if the puzzle involved only one ring, it would be extremely easy. Moreover, when faced with the problem of moving several rings, if you could move all but the largest ring to another peg, the largest ring could then be placed on the third peg, and then the problem would be to move the remaining rings on top of it. Using this observation, develop a recursive algorithm for solving the Towers of Hanoi puzzle for an arbitrary number of rings.

Short answer

Use recursion to move n-1 rings, move the nth ring, then move the n-1 rings again.

Step-by-step solution

Understanding the Problem

The Towers of Hanoi consists of moving a stack of rings from one peg to another peg. Rings must be moved one at a time and a larger ring cannot be placed on top of a smaller one.

Analysis of Problem with One Ring

If there is only one ring, the solution is straightforward: move it directly to the destination peg.

Recursive Approach Introduction

To solve it for multiple rings recursively: if we can move the top n-1 rings to an auxiliary peg, the nth ring can be moved to the destination peg. Then, move the n-1 rings on top of the nth ring on the destination peg.

Move n-1 Rings to Auxiliary Peg

Use the destination peg as an auxiliary peg to move the top n-1 rings from the source peg.

Move the nth Ring

Move the largest (nth) ring directly from the source peg to the destination peg.

Move n-1 Rings to Destination Peg

Finally, move the n-1 rings from the auxiliary peg to the destination peg, using the source peg as an auxiliary peg.

Recursive Pseudocode

Define a function `move(n, source, destination, auxiliary)` where: - If n == 1, move the ring directly from source to destination. - If n > 1, recursively call: 1. `move(n-1, source, auxiliary, destination)` to shift n-1 rings to the auxiliary peg. 2. Move the nth ring from the source to the destination. 3. `move(n-1, auxiliary, destination, source)` to shift n-1 rings onto the nth ring on the destination peg.

Key concepts

Towers of Hanoi

The Towers of Hanoi puzzle is a fascinating and classical exercise in understanding recursion. It involves three pegs: a source peg, an auxiliary peg, and a destination peg, along with a set of rings or disks that vary in size.
These rings are initially stacked in order on one peg, with the largest at the bottom and the smallest on top. The objective is simple, yet challenging: move the entire stack to another peg, adhering to strict rules.

• Only one ring can be moved at a time.
• A larger ring cannot be placed atop a smaller ring.

The challenge increases as the number of rings grows, and it intriguingly exemplifies the power of recursive thinking to solve complex problems step by step.

problem solving

Problem solving in the context of Towers of Hanoi requires breaking down a seemingly difficult task into manageable steps. Initially, it might appear daunting with several rings to move, but like many problem-solving scenarios, the key lies in simplification.
Start by understanding the nature of the problem with just a single ring: it is simply a matter of moving that one ring directly to the destination peg.
Once this is clear, the challenge is to develop a strategy for multiple rings.

• Visualize moving a stack of n-1 rings to an auxiliary peg.
• Shift the largest ring to the destination peg.
• Finally, move the n-1 rings from the auxiliary peg to the destination peg, on top of the largest ring.

By viewing the problem recursively, the intimidating task becomes a series of straightforward maneuvers, demonstrating a key skill in algorithmic problem-solving.

algorithm design

Algorithm design in solving the Towers of Hanoi involves a structured approach to detail each step required to solve the puzzle.
The recursive algorithm developed for this solves the problem by breaking it down systematically using the following steps:

• Define a function `move(n, source, destination, auxiliary)`.
• Base case: If there is just one ring (n == 1), move it from the source peg to the destination peg.
• Recursive case: For more than one ring (n > 1):
  • Move the top n-1 rings from the source peg to the auxiliary peg.
  • Move the nth ring directly to the destination peg.
  • Move the n-1 rings from the auxiliary peg to the destination peg.

This recursive design not only solves the problem but elegantly demonstrates how complex tasks can be approached by building stepwise solutions.

data structures

Data structures play an important role in implementation of the Towers of Hanoi algorithm as they allow efficient data management and manipulation.
In the case of this puzzle, using stacks as data structures can simplify the process as each peg in the Towers of Hanoi can be represented as a stack.
Here are some insights:

• Each peg (or stack) allows for storage and access of rings in a last-in, first-out (LIFO) manner.
• As rings must be removed from the top, this neatly aligns with how stacks operate.
• By pushing and popping rings between these stack structures, adherence to the puzzle's rules is ensured.

Utilizing stacks to model the pegs provides a concrete means to apply the recursive algorithm, demonstrating the interplay between complex problem-solving and the strategic use of data structures.