Why Tower of Hanoi Problem in DSA Java?

The Big Idea

Discover how a simple rule can solve a complex puzzle step-by-step without confusion!

The Scenario

Imagine you have three pegs and a stack of different sized disks on one peg. Your goal is to move all disks to another peg, but you can only move one disk at a time and never place a bigger disk on top of a smaller one.

Trying to figure out the steps manually for many disks is like solving a tricky puzzle without clues.

The Problem

Manually planning each move becomes confusing and error-prone as the number of disks grows. You might forget a step or break the rules, causing frustration and wasted time.

It's hard to keep track of all the moves and ensure the rules are followed without a clear method.

The Solution

The Tower of Hanoi problem uses a simple, clear set of rules and recursion to break down the big problem into smaller, manageable steps.

This method guarantees the correct sequence of moves without guesswork, making the solution elegant and easy to follow even for many disks.

Before vs After

✗ Before

void moveDisks(int n, char from, char to, char aux) {
  // Manually list each move for small n
  System.out.println("Move disk 1 from " + from + " to " + aux);
  System.out.println("Move disk 2 from " + from + " to " + to);
  System.out.println("Move disk 1 from " + aux + " to " + to);
}

✓ After

void moveDisks(int n, char from, char to, char aux) {
  if (n == 1) {
    System.out.println("Move disk 1 from " + from + " to " + to);
    return;
  }
  moveDisks(n - 1, from, aux, to);
  System.out.println("Move disk " + n + " from " + from + " to " + to);
  moveDisks(n - 1, aux, to, from);
}

What It Enables

This concept enables solving complex puzzles step-by-step with a clear, repeatable method that scales easily.

Real Life Example

It's like organizing a stack of plates from one shelf to another without breaking or mixing their order, using a helper shelf to hold plates temporarily.

Key Takeaways

Manual planning is confusing and error-prone for many disks.

Recursion breaks the problem into smaller, manageable steps.

The method guarantees a correct and efficient solution.

Practice

1. What is the minimum number of moves required to solve the Tower of Hanoi problem with 3 disks?

easy

A. 9

B. 6

C. 8

D. 7

Solution

1. Step 1: Understand the formula for minimum moves

   The minimum moves required for n disks is 2^n - 1.

2. Step 2: Calculate for 3 disks

   For 3 disks, minimum moves = 2^3 - 1 = 8 - 1 = 7.

3. Final Answer:

   7 -> Option D

4. Quick Check:

   2^3 - 1 = 7 [OK]

Hint: Use formula 2^n - 1 for minimum moves [OK]

Common Mistakes:

• Using n instead of 2^n - 1
• Calculating 2*n - 1 instead of 2^n - 1
• Confusing number of disks with moves

2. Which of the following is the correct base case condition in a recursive Tower of Hanoi function in Java?

easy

A. if (n < 0) return;

B. if (n == 1) return;

C. if (n == 0) return;

D. if (n > 1) return;

Solution

1. Step 1: Identify base case for recursion

   Base case stops recursion when no disks are left to move, i.e., n == 0.

2. Step 2: Check each option

   if (n == 0) return; correctly returns when n == 0, stopping recursion.

3. Final Answer:

   if (n == 0) return; -> Option C

4. Quick Check:

   Base case is n == 0 [OK]

Hint: Base case stops recursion at zero disks [OK]

Common Mistakes:

• Using n == 1 as base case which causes extra calls
• Using negative values as base case
• Returning when n > 1 which stops too early

3. What will be the output sequence of moves for Tower of Hanoi with 2 disks from source A to destination C using auxiliary B?

medium

A. Move disk 1 from A to B\nMove disk 2 from A to C\nMove disk 1 from B to C

B. Move disk 2 from A to B\nMove disk 1 from A to C\nMove disk 2 from B to C

C. Move disk 1 from A to C\nMove disk 2 from A to B\nMove disk 1 from C to B

D. Move disk 2 from A to C\nMove disk 1 from B to A\nMove disk 2 from C to B

Solution

1. Step 1: Understand move sequence for 2 disks

   Move smaller disk 1 from source A to auxiliary B.

2. Step 2: Move larger disk 2 from source A to destination C, then move disk 1 from B to C

   This completes the transfer following rules.

3. Final Answer:

   Move disk 1 from A to B\nMove disk 2 from A to C\nMove disk 1 from B to C -> Option A

4. Quick Check:

   Correct move order for 2 disks [OK]

Hint: Move smaller disk first to auxiliary, then largest to destination [OK]

Common Mistakes:

• Swapping disk numbers in moves
• Moving larger disk before smaller disk
• Incorrect order of moves

4. Identify the error in this Tower of Hanoi recursive function snippet:

void towerOfHanoi(int n, char from, char to, char aux) {
  if (n == 1) {
    System.out.println("Move disk 1 from " + from + " to " + to);
    return;
  }
  towerOfHanoi(n - 1, from, aux, to);
  System.out.println("Move disk " + n + " from " + from + " to " + to);
  towerOfHanoi(n - 1, aux, to, from);
}

medium

A. Base case should be if (n == 0) instead of if (n == 1)

B. There is no error; the function is correct

C. The print statement should be before the first recursive call

D. The order of parameters in recursive calls is incorrect

Solution

1. Step 1: Analyze base case correctness

   Base case n == 1 means move the single disk directly, which is correct.

2. Step 2: Check recursive calls and print order

   First move n-1 disks from source to auxiliary, then move nth disk, then move n-1 disks from auxiliary to destination. This matches the Tower of Hanoi logic.

3. Final Answer:

   There is no error; the function is correct -> Option B

4. Quick Check:

   Function matches Tower of Hanoi recursion [OK]

Hint: Check base case and recursive call order carefully [OK]

Common Mistakes:

• Changing base case to n == 0 causing infinite recursion
• Swapping parameters in recursive calls
• Moving print statement incorrectly

5. If the Tower of Hanoi problem has 4 disks, what is the total number of moves and what is the move sequence for the first two moves from source A to destination C using auxiliary B?

hard

A. Total moves: 15\nFirst move: Move disk 1 from A to B\nSecond move: Move disk 2 from A to C

B. Total moves: 14\nFirst move: Move disk 1 from A to B\nSecond move: Move disk 2 from B to C

C. Total moves: 16\nFirst move: Move disk 1 from A to B\nSecond move: Move disk 2 from A to C

D. Total moves: 15\nFirst move: Move disk 1 from A to C\nSecond move: Move disk 2 from A to B

Solution

1. Step 1: Calculate total moves for 4 disks

   Total moves = 2^4 - 1 = 16 - 1 = 15 moves.

2. Step 2: Determine first two moves in sequence

   First move: Move smallest disk 1 from A to B (auxiliary). Second move: Move disk 2 from A to C (destination).

3. Final Answer:

   Total moves: 15\nFirst move: Move disk 1 from A to B\nSecond move: Move disk 2 from A to C -> Option A

4. Quick Check:

   2^4 - 1 = 15 and correct first moves [OK]

Hint: Use formula for moves and remember smallest disk moves first [OK]

Common Mistakes:

• Incorrect total moves calculation
• Swapping first two moves
• Confusing auxiliary and destination rods