Solving the Tower of Hanoi Problem Using Iteration Instead of Recursion

The classic Tower of Hanoi puzzle presents an intriguing challenge that can be solved using both recursive and iterative approaches. While the recursive solution is elegant and straightforward, the iterative solution offers a unique way to solve the problem. In this article, we explore the iterative approach to solve the Tower of Hanoi puzzle and provide a detailed explanation along with a Python implementation.

Introduction to the Tower of Hanoi Puzzle

The Tower of Hanoi is a mathematical puzzle consisting of three pegs and a number of disks of different diameters, which can slide onto any peg. The puzzle starts with the disks in a neat stack in ascending order of size on one peg, the smallest at the top, thus making a conical shape. The goal of the puzzle is to move the entire stack to another peg, obeying the following simple rules:

Only one disk can be moved at a time. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty peg. No disk may be placed on top of a smaller disk.

Traditionally, the recursive solution is more intuitive and easier to implement. However, the iterative solution provides a fascinating alternative that adheres strictly to the rules of the game without relying on recursion.

Iterative Approach to Solve the Tower of Hanoi Puzzle

The iterative solution to the Tower of Hanoi puzzle is derived from the properties of the game. Here's a step-by-step explanation of how to solve it:

Number of Disks and Moves

Let n represent the number of disks:

1. The total number of moves required to solve the problem is 2n - 1.

Rules and Steps for the Iterative Solution

The rules for the iterative solution are:

Always move the smallest available disk. Follow the rules of the game.

Steps for the iterative solution:

Identify the Pegs: Label the pegs A, B, and C. Determine the Move Order: Depending on whether n is even or odd, the order of the moves changes: For odd n, the order is: A to C, A to B, B to C. For even n, the order is: A to B, A to C, B to C. Perform Moves: Use a loop to perform 2n - 1 moves following the identified move order.

Example Code in Python

Here is a simple implementation of the iterative solution in Python:

import math
def tower_of_hanoi_iterative(n):
    # Calculate the total number of moves
    total_moves  int(math.pow(2, n) - 1)
    # Peg labels
    pegs  ['A', 'B', 'C']
    # If n is even, swap the destination pegs
    if n % 2  0:
        pegs[1], pegs[2]  pegs[2], pegs[1]
    # Perform the moves
    for move in range(total_moves):
        from_peg  move  3  # A  0, B  1, C  2
        to_peg  (move  2) | (move  1)  1  # Next peg
        print(f"Move disk from {pegs[from_peg]} to {pegs[to_peg]}")
# Example usage
tower_of_hanoi_iterative(3)

The function calculates the total number of moves and sets up the peg labels. It checks if the number of disks is even and adjusts the order of pegs accordingly. The loop iterates through the total number of moves, determining which disk to move based on the current move number.

Comparison with the Recursive Approach

The recursive solution is more intuitive and easier to understand. The iterative solution, although more complex, offers a unique perspective on the problem. The iterative approach simulates the recursive process and adheres strictly to the rules of the game, providing a challenge for those interested in exploring alternative solutions.

Conclusion

In conclusion, solving the Tower of Hanoi problem using an iterative approach is not only possible but also provides an interesting alternative to the traditional recursive solution. The iterative method adheres to the rules of the game while using a loop-based approach, making it a fascinating topic for puzzle enthusiasts and algorithm developers.