Why Tower of Hanoi Problem in DSA C#?

The Big Idea

What if a simple rule could solve a puzzle that seems impossible to plan by hand?

The Scenario

Imagine you have three pegs and a stack of different-sized disks on one peg. You want 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 gets confusing and slow as the number of disks grows. You might forget rules or make mistakes, causing you to redo steps.

This makes the task frustrating and error-prone, especially with more disks.

The Solution

The Tower of Hanoi problem uses a simple set of rules and recursion to break the big problem into smaller ones. It tells you exactly which disk to move and when, without guessing.

This method saves time and avoids mistakes by following a clear pattern.

Before vs After

✗ Before

void MoveDisk(int disk, char fromPeg, char toPeg) {
  // Manually write each move
  Console.WriteLine($"Move disk {disk} from {fromPeg} to {toPeg}");
}

// For 3 disks, manually call MoveDisk many times

✓ After

void SolveHanoi(int numberOfDisks, char fromPeg, char toPeg, char auxPeg) {
  if (numberOfDisks == 1) {
    Console.WriteLine($"Move disk 1 from {fromPeg} to {toPeg}");
    return;
  }
  SolveHanoi(numberOfDisks - 1, fromPeg, auxPeg, toPeg);
  Console.WriteLine($"Move disk {numberOfDisks} from {fromPeg} to {toPeg}");
  SolveHanoi(numberOfDisks - 1, auxPeg, toPeg, fromPeg);
}

What It Enables

This problem-solving approach enables you to solve complex puzzles step-by-step with clear instructions, no matter how many disks you have.

Real Life Example

Think of organizing stacked boxes in a warehouse where you can only move one box at a time and must keep bigger boxes below smaller ones. The Tower of Hanoi method helps plan the moves efficiently.

Key Takeaways

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

Recursion breaks the problem into smaller, manageable steps.

The method gives a clear, repeatable way to solve the puzzle.

Practice

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

easy

A. 6

B. 7

C. 8

D. 9

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 B

4. Quick Check:

   2^3 - 1 = 7 [OK]

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

Common Mistakes:

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

2. Which of the following is the correct C# method signature for a recursive Tower of Hanoi function?

easy

A. void TowerOfHanoi(int n, string fromRod, string toRod)

B. int TowerOfHanoi(int n, char fromRod, char toRod, char auxRod)

C. int TowerOfHanoi(int n, string fromRod, string toRod, string auxRod)

D. void TowerOfHanoi(int n, char fromRod, char toRod, char auxRod)

Solution

1. Step 1: Identify correct parameter types

   The Tower of Hanoi function typically uses integer for disk count and characters for rods.

2. Step 2: Check return type and parameters

   The function usually returns void and takes 3 rods as char parameters plus disk count as int.

3. Final Answer:

   void TowerOfHanoi(int n, char fromRod, char toRod, char auxRod) -> Option D

4. Quick Check:

   Correct signature uses void and char rods [OK]

Hint: Use void return and char for rods in signature [OK]

Common Mistakes:

• Using string instead of char for rods
• Returning int instead of void
• Missing auxiliary rod parameter

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

medium

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

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

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

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

Solution

1. Step 1: Understand the recursive move sequence for 2 disks

   Move smaller disk 1 to auxiliary rod B, then move disk 2 to target rod C, then move disk 1 from B to C.

2. Step 2: Verify the move order

   Sequence is: 1 from A to B, 2 from A to C, 1 from B to C.

3. Final Answer:

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

4. Quick Check:

   Correct move order for 2 disks [OK]

Hint: For 2 disks, move smaller disk first to aux rod [OK]

Common Mistakes:

• Swapping disk numbers in moves
• Confusing auxiliary and target rods
• Reversing move order

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

void Hanoi(int n, char from, char to, char aux) {
  if (n == 0) return;
  Hanoi(n - 1, from, aux, to);
  Console.WriteLine($"Move disk {n} from {from} to {to}");
  Hanoi(n - 1, aux, to, from);
}

medium

A. Base case should be if (n == 1) return;

B. Recursive calls have wrong rod order in parameters

C. No error, code is correct

D. Console.WriteLine should print disk n-1 instead of n

Solution

1. Step 1: Check base case correctness

   Base case n == 0 means no disks to move, which is correct to stop recursion.

2. Step 2: Verify recursive calls and print statement

   Recursive calls correctly move n-1 disks between rods, and print statement moves disk n from 'from' to 'to'.

3. Final Answer:

   No error, code is correct -> Option C

4. Quick Check:

   Base case and recursion are correctly implemented [OK]

Hint: Base case n==0 is correct to stop recursion [OK]

Common Mistakes:

• Changing base case to n==1 incorrectly
• Mixing up rod parameters in recursive calls
• Printing wrong disk number

5. You want to solve Tower of Hanoi for 4 disks but print only the moves where disk 1 is moved. How many such moves will be printed?

hard

A. 8

B. 15

C. 7

D. 4

Solution

1. Step 1: Calculate total moves for 4 disks

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

2. Step 2: Count moves involving disk 1

   Disk 1 moves every alternate move, so number of moves = (total moves + 1) / 2 = (15 + 1)/2 = 8.

3. Final Answer:

   8 -> Option A

4. Quick Check:

   Disk 1 moves half the times plus one [OK]

Hint: Disk 1 moves on every alternate move, count half plus one [OK]

Common Mistakes:

• Assuming disk 1 moves equal to total moves
• Using total moves minus one
• Confusing disk 1 moves with total disks