Question

The Towers of Hanoi is a classic puzzle that consists of 3 towers, and \(n\) disks of different sizes that can slide onto any tower. The puzzle starts with the disks sorted in ascending order of size from top to bottom. It has the following constraints: a. Only one disk can be moved at a time. b. A disk can only be moved from the top of one tower to another tower. c. A disk can only be placed on top of a larger disk. Write a program to move the disks from the first tower to the last using stacks.

Short answer

Solve the Towers of Hanoi with recursion using the function `solveHanoi(n, source, target, auxiliary)`.

Step-by-step solution

Understand the Problem

The Towers of Hanoi problem involves moving all disks from the first tower to the last tower, following specific rules: only one disk can be moved at a time, disks must be moved from the top of one tower to another, and no disk can be placed on top of a smaller disk.

Recognize the Recursive Solution

The problem is commonly solved using recursion. The process involves moving the top n-1 disks from the source tower to an auxiliary tower, moving the largest disk directly to the target tower, and finally moving the n-1 disks from the auxiliary tower to the target tower.

Define the Recursive Function

In a programming context, the recursive function can be defined as `solveHanoi(n, source, target, auxiliary)`, where `n` is the number of disks, `source` is the initial tower, `target` is the final tower, and `auxiliary` is the temporary tower used during the process.

Base Case for Recursion

The base case for the recursion is when there is only one disk to move, i.e., if `n == 1`, simply move the disk from the source tower to the target tower and return.

Recursive Case for Multiple Disks

For more than one disk, recursively solve three subproblems: 1. Move the top `n-1` disks from the source tower to the auxiliary tower. 2. Move the nth disk (largest) directly from the source tower to the target tower. 3. Move the `n-1` disks from the auxiliary tower to the target tower.

Implement the Program

Using a programming language like Python, implement the recursive function. The program inputs the number of disks `n` and outputs the sequence of moves needed to solve the puzzle.

Key concepts

Towers of Hanoi

The Towers of Hanoi is an intriguing puzzle widely used in computer science education. It helps people understand recursion and algorithmic thinking.
It consists of three rods and a number of disks of different sizes that can slide onto any rod. The puzzle begins with all disks stacked in descending order of size on one rod.
The main objective is to move the entire stack to another rod, following these rules:

• Only one disk is allowed to be moved at a time.
• A move consists of taking a disk from the top of one rod and placing it on another rod.
• No larger disk can be placed on top of a smaller disk.

These simple rules create a complex problem that requires strategic planning and logical reasoning.

Recursive Function

A recursive function is a function that calls itself in order to solve a problem. This is particularly useful in problems that can be broken down into smaller, similar sub-problems.
In the Towers of Hanoi puzzle, recursion is a natural choice. By framing each move in terms of previous moves, you can break down the problem into smaller and easier pieces.
The recursive solution involves moving smaller subsets of disks between the rods before tackling the largest disk, following a specific sequence of steps. The key here is to use an auxiliary rod to help swap the disks between the source and target rods.
This self-referential nature allows the algorithm to effectively manage operations, eventually solving the puzzle.

Algorithm Design

Algorithm design involves creating a step-by-step approach to solve a problem efficiently. For the Towers of Hanoi, the essence of the algorithm is casting this problem into a recursive framework.
You define a function `solveHanoi(n, source, target, auxiliary)` where:

• `n` is the number of disks.
• `source` is the starting rod.
• `target` is the rod where disks should eventually move.
• `auxiliary` is the helper rod used in the process.

The base case is when there's only one disk to move. The recursive case involves solving the problem for smaller numbers of disks, eventually leading to a complete solution.
Well-designed algorithms explore the logic behind the problem, allowing for predictable results with an efficient number of steps.

Problem Solving in Computer Science

Problem solving in computer science often involves breaking complex problems into simpler ones. The Towers of Hanoi is a perfect example of this process.
Through recursion, complex tasks are simplified into smaller, manageable actions. Understanding such foundational concepts empowers programmers to tackle advanced problems with confidence.
Using recursive functions to address a problem like Towers of Hanoi demonstrates a mastery of problem decomposition and solution composition. It teaches programmers to identify patterns, optimize processes, and ultimately learn how to solve a wide variety of computational challenges.
Developing such problem-solving skills is invaluable in various domains of computer science, from designing algorithms to optimizing systems.