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Solution Approach

Solution Overview

There are 1 main approaches to solve this problem:

Recursive Approach - Complex approach

Approach 1: Recursive Approach

The Tower of Hanoi puzzle is a classic example of a problem that can be solved elegantly using recursion. The key insight is to break down the problem into smaller subproblems.

To move n disks from the source rod to the target rod:

Move n-1 disks from the source rod to the auxiliary rod.
Move the largest disk (disk n) from the source rod to the target rod.
Move the n-1 disks from the auxiliary rod to the target rod.

Each of these steps can be solved recursively, with the base case being moving a single disk directly from one rod to another.

Algorithm:

Define a recursive function towerOfHanoi(n, source, auxiliary, target) that takes the number of disks and the three rods as parameters.
Base case: If n = 1, print the move to transfer the single disk from source to target.
Recursive case: For n > 1, break down the problem into three steps:
1. Recursively move n-1 disks from source to auxiliary, using target as the temporary rod.
2. Move disk n from source to target (print this move).
3. Recursively move n-1 disks from auxiliary to target, using source as the temporary rod.

Time Complexity:

O(2^n)

The recurrence relation is T(n) = 2T(n-1) + 1, which resolves to O(2^n). This matches the fact that the minimum number of moves required is 2^n - 1.

Space Complexity:

O(n)

The recursion stack will have at most n frames, each using constant space.

Pseudocode

solution.pseudo

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function towerOfHanoi(n, source, auxiliary, target):
    // Base case: only one disk to move
    if n == 1:
        print("Move disk 1 from " + source + " to " + target)
        return
 
    // Step 1: Move n-1 disks from source to auxiliary
    towerOfHanoi(n - 1, source, target, auxiliary)
 
    // Step 2: Move the largest disk from source to target
    print("Move disk " + n + " from " + source + " to " + target)
 
    // Step 3: Move n-1 disks from auxiliary to target
    towerOfHanoi(n - 1, auxiliary, source, target)

String Problems

Palindrome Checker

Determine if a string reads the same forward and backward.

EasyString Manipulation

Message Encryption

Reverse a string to create a simple encryption system.

EasyString Manipulation

Word Puzzle Challenge

Check if two strings are anagrams of each other.

EasyString Manipulation

Learning Objective

Understand different approaches to solve the tower of hanoi puzzle solver problem and analyze their efficiency.

Solution Approaches

Approach 1: Recursive Approach

The Tower of Hanoi puzzle is a classic example of a problem that can be solved elegantly using recursion. The key insight is to break down the problem into smaller subproblems.

To move n disks from the source rod to the target rod:

Move n-1 disks from the source rod to the auxiliary rod.
Move the largest disk (disk n) from the source rod to the target rod.
Move the n-1 disks from the auxiliary rod to the target rod.

Each of these steps can be solved recursively, with the base case being moving a single disk directly from one rod to another.

Algorithm:

Define a recursive function towerOfHanoi(n, source, auxiliary, target) that takes the number of disks and the three rods as parameters.
Base case: If n = 1, print the move to transfer the single disk from source to target.
Recursive case: For n > 1, break down the problem into three steps:
1. Recursively move n-1 disks from source to auxiliary, using target as the temporary rod.
2. Move disk n from source to target (print this move).
3. Recursively move n-1 disks from auxiliary to target, using source as the temporary rod.

Complexity Analysis

Recursive Approach Approach

Time Complexity

O(2^n)

The recurrence relation is T(n) = 2T(n-1) + 1, which resolves to O(2^n). This matches the fact that the minimum number of moves required is 2^n - 1.

Space Complexity

O(n)

The recursion stack will have at most n frames, each using constant space.

Comparison:

The recursive approach is the most natural and elegant solution for the Tower of Hanoi puzzle. While iterative solutions exist, they are more complex and less intuitive. The recursive solution directly models the problem-solving strategy and clearly demonstrates the power of recursion for breaking down complex problems into simpler subproblems.

Pseudocode

Recursive Approach Approach

approach1.pseudo

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function towerOfHanoi(n, source, auxiliary, target):
    // Base case: only one disk to move
    if n == 1:
        print("Move disk 1 from " + source + " to " + target)
        return
 
    // Step 1: Move n-1 disks from source to auxiliary
    towerOfHanoi(n - 1, source, target, auxiliary)
 
    // Step 2: Move the largest disk from source to target
    print("Move disk " + n + " from " + source + " to " + target)
 
    // Step 3: Move n-1 disks from auxiliary to target
    towerOfHanoi(n - 1, auxiliary, source, target)

Problem Statement Implementation

Problem Solution Code

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Tower of Hanoi Puzzle Solver - Solution

Solution Approach

Solution Overview

There are 1 main approaches to solve this problem:

Recursive Approach - Complex approach

Approach 1: Recursive Approach

The Tower of Hanoi puzzle is a classic example of a problem that can be solved elegantly using recursion. The key insight is to break down the problem into smaller subproblems.

To move n disks from the source rod to the target rod:

Move n-1 disks from the source rod to the auxiliary rod.
Move the largest disk (disk n) from the source rod to the target rod.
Move the n-1 disks from the auxiliary rod to the target rod.

Each of these steps can be solved recursively, with the base case being moving a single disk directly from one rod to another.

Algorithm:

Define a recursive function towerOfHanoi(n, source, auxiliary, target) that takes the number of disks and the three rods as parameters.
Base case: If n = 1, print the move to transfer the single disk from source to target.
Recursive case: For n > 1, break down the problem into three steps:
1. Recursively move n-1 disks from source to auxiliary, using target as the temporary rod.
2. Move disk n from source to target (print this move).
3. Recursively move n-1 disks from auxiliary to target, using source as the temporary rod.

Time Complexity:

O(2^n)

The recurrence relation is T(n) = 2T(n-1) + 1, which resolves to O(2^n). This matches the fact that the minimum number of moves required is 2^n - 1.

Space Complexity:

O(n)

The recursion stack will have at most n frames, each using constant space.

Pseudocode

solution.pseudo

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function towerOfHanoi(n, source, auxiliary, target):
    // Base case: only one disk to move
    if n == 1:
        print("Move disk 1 from " + source + " to " + target)
        return
 
    // Step 1: Move n-1 disks from source to auxiliary
    towerOfHanoi(n - 1, source, target, auxiliary)
 
    // Step 2: Move the largest disk from source to target
    print("Move disk " + n + " from " + source + " to " + target)
 
    // Step 3: Move n-1 disks from auxiliary to target
    towerOfHanoi(n - 1, auxiliary, source, target)

String Problems

Palindrome Checker

Determine if a string reads the same forward and backward.

EasyString Manipulation

Message Encryption

Reverse a string to create a simple encryption system.

EasyString Manipulation

Word Puzzle Challenge

Check if two strings are anagrams of each other.

EasyString Manipulation

Learning Objective

Understand different approaches to solve the tower of hanoi puzzle solver problem and analyze their efficiency.

Solution Approaches

Approach 1: Recursive Approach

The Tower of Hanoi puzzle is a classic example of a problem that can be solved elegantly using recursion. The key insight is to break down the problem into smaller subproblems.

To move n disks from the source rod to the target rod:

Move n-1 disks from the source rod to the auxiliary rod.
Move the largest disk (disk n) from the source rod to the target rod.
Move the n-1 disks from the auxiliary rod to the target rod.

Each of these steps can be solved recursively, with the base case being moving a single disk directly from one rod to another.

Algorithm:

Define a recursive function towerOfHanoi(n, source, auxiliary, target) that takes the number of disks and the three rods as parameters.
Base case: If n = 1, print the move to transfer the single disk from source to target.
Recursive case: For n > 1, break down the problem into three steps:
1. Recursively move n-1 disks from source to auxiliary, using target as the temporary rod.
2. Move disk n from source to target (print this move).
3. Recursively move n-1 disks from auxiliary to target, using source as the temporary rod.

Complexity Analysis

Recursive Approach Approach

Time Complexity

O(2^n)

The recurrence relation is T(n) = 2T(n-1) + 1, which resolves to O(2^n). This matches the fact that the minimum number of moves required is 2^n - 1.

Space Complexity

O(n)

The recursion stack will have at most n frames, each using constant space.

Comparison:

The recursive approach is the most natural and elegant solution for the Tower of Hanoi puzzle. While iterative solutions exist, they are more complex and less intuitive. The recursive solution directly models the problem-solving strategy and clearly demonstrates the power of recursion for breaking down complex problems into simpler subproblems.

Pseudocode

Recursive Approach Approach

approach1.pseudo

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function towerOfHanoi(n, source, auxiliary, target):
    // Base case: only one disk to move
    if n == 1:
        print("Move disk 1 from " + source + " to " + target)
        return
 
    // Step 1: Move n-1 disks from source to auxiliary
    towerOfHanoi(n - 1, source, target, auxiliary)
 
    // Step 2: Move the largest disk from source to target
    print("Move disk " + n + " from " + source + " to " + target)
 
    // Step 3: Move n-1 disks from auxiliary to target
    towerOfHanoi(n - 1, auxiliary, source, target)

Problem Statement Implementation