Why Tower of Hanoi Problem in DSA Python?

The Big Idea

Discover how a simple rule can solve a seemingly impossible puzzle step by step!

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 gets confusing very fast as the number of disks grows.

The Problem

Manually planning each move is slow and easy to mess up. You might forget a step or break the rules by placing a bigger disk on a smaller one. The number of moves grows quickly, making it almost impossible to keep track 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 ones. This method tells you exactly which disk to move and when, so you never get stuck or confused.

Before vs After

✗ Before

def move_disks(n, source, target, auxiliary):
    # Try to move disks manually without a plan
    pass  # Too complex to write manually

✓ After

def move_disks(n, source, target, auxiliary):
    if n == 1:
        print(f"Move disk 1 from {source} to {target}")
    else:
        move_disks(n-1, source, auxiliary, target)
        print(f"Move disk {n} from {source} to {target}")
        move_disks(n-1, auxiliary, target, source)

What It Enables

This problem teaches how to solve complex tasks by breaking them into smaller, manageable steps using recursion.

Real Life Example

Moving a stack of plates from one shelf to another without breaking or mixing them up, by carefully moving one plate at a time following a clear plan.

Key Takeaways

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

Recursion breaks the problem into smaller, easier steps.

Following the recursive method guarantees a correct solution.

Practice

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

easy

A. 5

B. 7

C. 9

D. 6

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:

• 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 Python?

easy

A. if n == 0: return

B. if n < 0: return

C. if n > 1: call recursion

D. if n == 1: move disk

Solution

1. Step 1: Identify base case for recursion

   The base case stops recursion when only one disk is left, i.e., n == 1.

2. Step 2: Understand base case action

   When n == 1, the disk is moved directly from source to target.

3. Final Answer:

   if n == 1: move disk -> Option D

4. Quick Check:

   Base case is n == 1 [OK]

Hint: Base case is when only one disk remains (n == 1) [OK]

Common Mistakes:

• Using n == 0 as base case incorrectly
• Not defining a base case causing infinite recursion
• Using invalid conditions like n < 0

3. What will be the output of the following Python code snippet for Tower of Hanoi with 2 disks?

def hanoi(n, source, target, auxiliary):
    if n == 0:
        return
    hanoi(n-1, source, auxiliary, target)
    print(f"Move disk {n} from {source} to {target}")
    hanoi(n-1, auxiliary, target, source)

hanoi(2, 'A', 'C', 'B')

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

Solution

1. Step 1: Trace recursive calls for n=2

   First, move disk 1 from A to B, then disk 2 from A to C, then disk 1 from B to C.

2. Step 2: Verify print statements order

   Prints match the moves: disk 1 A->B, disk 2 A->C, disk 1 B->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 sequence for 2 disks [OK]

Hint: Follow recursive calls step-by-step [OK]

Common Mistakes:

• Mixing source and auxiliary rods
• Printing moves in wrong order
• Confusing disk numbers

4. Identify the error in this Tower of Hanoi recursive function and select the correct fix:

def hanoi(n, source, target, auxiliary):
    if n == 1:
        print(f"Move disk {n} from {source} to {target}")
    else:
        hanoi(n-1, source, target, auxiliary)
        print(f"Move disk {n} from {source} to {target}")
        hanoi(n-1, auxiliary, target, source)

medium

A. Change first recursive call to hanoi(n-1, source, auxiliary, target)

B. Change base case to if n == 0

C. Swap print statements order

D. Remove the else block

Solution

1. Step 1: Analyze recursive calls

   The first recursive call incorrectly moves disks from source to target instead of auxiliary.

2. Step 2: Correct the first recursive call

   It should be hanoi(n-1, source, auxiliary, target) to move disks to auxiliary rod first.

3. Final Answer:

   Change first recursive call to hanoi(n-1, source, auxiliary, target) -> Option A

4. Quick Check:

   Correct recursive call order fixes the error [OK]

Hint: First recursive call moves disks to auxiliary rod [OK]

Common Mistakes:

• Calling recursion with wrong target rod
• Changing base case incorrectly
• Swapping print statements unnecessarily

5. You want to solve Tower of Hanoi for 4 disks but only have 3 rods. What is the minimum number of moves required, and why can't it be less than that?

hard

A. 14 moves, because the largest disk can be moved directly

B. 16 moves, because one extra move is needed for 4 disks

C. 15 moves, because 2^4 - 1 is the minimum for 3 rods

D. 7 moves, because only half the disks need moving

Solution

1. Step 1: Apply minimum moves formula for 3 rods

   Minimum moves for n disks with 3 rods is 2^n - 1, so for 4 disks: 2^4 - 1 = 15.

2. Step 2: Understand why moves can't be less

   With 3 rods, no shortcut exists; all disks must be moved following rules, so 15 is minimum.

3. Final Answer:

   15 moves, because 2^4 - 1 is the minimum for 3 rods -> Option C

4. Quick Check:

   Minimum moves = 2^n - 1 for 3 rods [OK]

Hint: Use 2^n - 1 formula for 3 rods minimum moves [OK]

Common Mistakes:

• Assuming fewer moves possible with 3 rods
• Adding extra moves unnecessarily
• Confusing number of rods with moves