Question

In this chapter, you studied functions that can be easily implemented both recursively and iteratively.. In this exercise, we present a problem whose recursive solution demonstrates the elegance of recursion, and whose iterative solution may not be as apparent. The Towers of Hanoi is one of the most famous classic problems every budding computer scientist must grapple with. Legend has it that in a temple in the Far East, priests are attempting to move a stack of golden disks from one diamond peg to another (Fig. 6.35). The initial stack has 64 disks threaded onto one peg and arranged from bottom to top by decreasing size. The priests are attempting to move the stack from one peg to another under the constraints that exactly one disk is moved at a time and at no time may a larger disk be placed above a smaller disk. Three pegs are provided, one being used for temporarily holding disks. Supposedly, the world will end when the priests complete their task, so there is little incentive for us to facilitate their efforts. Let’s assume that the priests are attempting to move the disks from peg 1 to peg 3. We wish to develop an algorithm that prints the precise sequence of peg-to-peg disk transfers. If we were to approach this problem with conventional methods, we would rapidly find ourselves hopelessly knotted up in managing the disks. Instead, attacking this problem with recursion in mind allows the steps to be simple. Moving n disks can be viewed in terms of moving only n – 1 disks (hence, the recursion), as follows: a) Move \(n-1\) disks from peg 1 to peg \(2,\) using peg 3 as a temporary holding area. b) Move the last disk (the largest) from peg 1 to peg 3. c) Move the \(n-1\) disks from peg 2 to peg \(3,\) using peg 1 as a temporary holding area. The process ends when the last task involves moving \(n=1\) disk (i.e., the base case). This task is accomplished by simply moving the disk, without the need for a temporary holding area. Write a program to solve the Towers of Hanoi problem. Use a recursive function with four parameters: a) The number of disks to be moved b) The peg on which these disks are initially threaded c) The peg to which this stack of disks is to be moved d) The peg to be used as a temporary holding area Display the precise instructions for moving the disks from the starting peg to the destination peg. To move a stack of three disks from peg 1 to peg 3, the program displays the following moves: \(1 \rightarrow 3\) (This means move one disk from peg 1 to peg \(3 .\) ) \\[ \begin{array}{l} 1 \rightarrow 2 \\ 3 \rightarrow 2 \\ 1 \rightarrow 3 \\ 2 \rightarrow 1 \\ 2 \rightarrow 3 \\ 1 \rightarrow 3 \end{array} \\]

Short answer

To solve the problem, create a recursive function with a base case that directly moves the disk from the start to the target peg, and a recursive case that moves the stack of `n - 1` disks to the temporary peg, then moves the largest disk, and finally moves the `n - 1` stack to the target peg.

Step-by-step solution

- Understand the Problem

First, comprehend the recursive nature of the Towers of Hanoi problem. Realize that you can move a stack of disks by moving all but the lowest one to a temporary peg, then move the lowest disk, and finally move the stack from the temporary peg to the target peg.

- Write a Recursive Function

Create a recursive function `hanoi` with four parameters: number of disks `n`, the start peg `start`, the target peg `target`, and the temporary peg `temp`. The function will print the moves necessary to move `n` disks from the `start` peg to the `target` peg.

- Define the Base Case

Define the base case for the recursive function: if `n == 1`, print the move directly from the start peg to the target peg. This is the simplest scenario with no need for a temporary peg.

- Define the Recursive Case

Define the recursive case within the function for when `n > 1`. First, recursively move `n - 1` disks from the start peg to the temporary peg. Next, print the move of the remaining disk from the start peg to the target peg. Lastly, recursively move the `n - 1` disks from the temporary peg to the target peg.

- Testing the Function

Test the `hanoi` function with a small number of disks, such as 3, to ensure the output reflects the expected sequence of moves.

Key concepts

Recursion in Programming

Recursion is a powerful concept in programming where a function calls itself to solve a problem. It's particularly useful in scenarios where a task can be divided into similar subtasks. The beauty of recursive solutions lies in their simplicity and elegance, often leading to more intuitive code.

Take the Towers of Hanoi problem, for example. It perfectly illustrates how recursion simplifies a complex issue. By breaking down the problem of moving a stack of disks into smaller stages—each involving moving fewer disks—the solution becomes manageable and clear. Recursive strategies frequently involve a 'base case' to terminate the recursion. This allows our function to stop calling itself endlessly and, in essence, begins the process of solving the problem in a reverse order from the smallest piece back up to the original task.

Recursive Functions

In C++, recursive functions are implemented by a function calling itself with modified parameters that move towards a base case. For the Towers of Hanoi problem, our recursive function has four parameters: the number of disks, the start peg, the target peg, and a temporary holding peg.

Understanding a recursive function's structure is crucial. The base case acts as an anchor, providing a simple stop point for the recursion. Above the base case, the recursive calls work their way up to the initial complex problem we started with. It's like unraveling a stack of nested boxes, where each box corresponds to a recursive call that brings us one step closer to the solution.

Algorithmic Problem Solving

Algorithmic problem solving involves creating step-by-step methods for solving problems. The Tower of Hanoi algorithm, through a recursive approach, utilizes this by breaking the problem into smaller, more manageable parts—moving fewer disks amongst pegs—until reaching the state where only one disk is to be moved (the simplest form of the problem).

One of the major perks of algorithmic problem solving is the ability to test and verify each part of the process. In the case of the Towers of Hanoi, each recursive call can be tested to ensure that the procedure for moving the disks follows the rules and constraints given in the problem. This step-by-step verification process makes debugging simpler and helps in understanding the fundamentals of both the problem and the solution.

C++ Programming Exercises

Practical exercises in C++, like solving the Towers of Hanoi, reinforce key programming concepts such as recursion and control structures. For students tackling programming exercises, transforming theoretical knowledge into practice is essential. Writing a C++ program to solve the Towers of Hanoi fosters a deeper understanding of recursive functions and the mechanics behind them.

Challenges like these encourage students to think critically about how to structure a program and systematically tackle the given task. Plus, successfully solving complex problems through code can be incredibly rewarding and is indicative of a strong understanding of the underlying principles of computer science.